Laboratoire de Physique Statistique, École
Normale Supérieure, Unité de Recherche 8550 associée
au Centre National de la Recherche Scientifique et aux
Universités Paris VI et VII, 75231 Paris, France
Cantilevers and optical tweezers are widely used
for micromanipulating cells or biomolecules for measuring their
mechanical properties. However, they do not allow easy rotary motion
and can sometimes damage the handled material. We present here a system of magnetic tweezers that overcomes those drawbacks while retaining most of the previous dynamometers properties. Electromagnets are coupled to a microscope-based particle tracking system through a
digital feedback loop. Magnetic beads are first trapped in a potential
well of stiffness ~10
7 N/m. Thus, they can be
manipulated in three dimensions at a speed of ~10 µm/s and rotated
along the optical axis at a frequency of 10 Hz. In addition, our
apparatus can work as a dynamometer relying on either usual calibration
against the viscous drag or complete calibration using Brownian
fluctuations. By stretching a DNA molecule between a magnetic particle
and a glass surface, we applied and measured vertical forces ranging
from 50 fN to 20 pN. Similarly, nearly horizontal forces up to 5 pN
were obtained. From those experiments, we conclude that magnetic
tweezers represent a low-cost and biocompatible setup that could become
a suitable alternative to the other available micromanipulators.
 |
LIST OF SYMBOLS |
| Au |
(NA1) the factor between the force and the
current flowing in the coils (Defined in Eq. 2) |
 |
(nondimensional) the proportional feedback factor in the
digital model (Defined in Eq. 11) |
| Bu |
(µm s1A1) the proportional
factor between the velocity of a bead and the associated driving
current (Defined in Eq. 19) |
 |
(nondimensional) the ratio between the integral and the
proportional feedback factors (Defined in Eq. 17) |
| Cu |
(nondimensional) the correction signal in feedback loop
(Defined in Eq. 1) |
| Cr(f) |
(nondimensional) the camera filtering correction due to its
finite integration time (Defined in Eq. 32) |
| D |
(µm2s1) the bead diffusion
coefficient (Defined in Eq. 9) |
 t |
(s) the time interval between two video frames (40 ms)
(Defined in Eq. 12) |
| ti |
(s) the time integration of the video camera (0.97 t) (Defined in Eq. 32) |
 |
(nondimensional) the delay in the feedback loop (Defined in
Eq. 11) |
| T |
(s) the integration time of the signal (Defined after Eq. 10) |
 |
(poise) the fluid viscosity (Defined in Eq. 7) |
| Fu |
(N) the force acting on the bead (Defined in Eq. 2) |
| FL |
(N) the random Langevin force responsible for the Brownian
motion (Defined in Eq. 7) |
| fc |
(Hz) the cutoff frequency of the bead attached to the
molecule system, fc = ku/(2  r) (Defined in the
paragraph just after Eq. 8). |
| fs |
(Hz) the sampling frequency of the camera (25 Hz here),
fs = 1/t (Defined in Eq. 31) |
| fL |
(Hz) the smallest usable frequency,
fL = 1/T (Defined after Eq. 10) |
| |
(nondimensional) the viscous coefficient of a sphere,
usually 6 (Defined in Eq. 7) |
| x,y |
(nondimensional) the viscous coefficient of a sphere moving
parallel to a sidewall (Defined in Eq. 30) |
| z |
(nondimensional) the viscous coefficient of a sphere moving
perpendicularly to a sidewall (Defined in Eq. 30) |
| Iu |
(A) the coil driving current (Defined in Eq. 1) |
| Ium |
(A) the maximum current set in one direction (Defined in
Characterization of the Active Tweezers). |
| I0 |
(A) the minimal value of Iz that
lift the magnetic particle. (Defined in Design of the Apparatus). |
| ku |
(Nm1) the stiffness of the tweezers (Defined
in Eq. 2) |
| Ku |
(µm1) the integral coefficient in the
feedback loop (Defined in Eq. 1) |
L (f) |
(µm2Hz1) the Langevin force
noise density in Fourier space (Defined in Eq. 14) |
| Pu |
(µm1) the proportional coefficient in the
feedback loop (Defined in Eq. 1) |
| r |
(µm) the bead radius (Defined in Eq. 7) |
| Vu |
(µm s1) the bead velocity (Defined in Eq. 9) |
| |
INTRODUCTION |
During the last ten years, single biomolecule
micromanipulations have revolutionized the field of biophysics
(Bensimon, 1996
; Bustamante et al.,
2000), allowing the biophysicists 1) to measure the elastic
behavior of biopolymers such as actin (Kishino and Yanagida,
1988), titin (Kellermayer et al., 1997
Carrion-Vazquez et al, 1999), or DNA (Cluzel et
al., 1996; Strick et al., 1996); 2) to determine
the tensil strength of single ligand/receptor bond (Florin et
al., 1994; Merkel et al., 1999); 3) to
investigate the micromechanics of molecular motors such as kinesin
(Block et al., 1990) and myosin (Ishijima et al.,
1991; Finer et al., 1994); 4) to follow in real
time the activity of single proteins such as polymerases (Yin et
al., 1995; Maier et al., 2000; Wuite et
al., 2000b); and even 5) to observe single enzymatic cycles of
individual enzymes (Noji et al., 1997; Strick et
al., 2000).
Micromanipulation implies monitoring of forces at the molecular scale.
In biology, at the single-molecule level, the characteristic energy is
given by the hydrolysis of ATP (20 kT i.e., 80 pN·nm) and
the characteristic size by the diameter of a protein (a few nanometers). The resulting forces that biophysicists must be able to
measure and to produce while studying those objects are therefore in
the range of hundreds of femtonewtons to tens of piconewtons. In most
of the previously mentioned studies, the biomolecule is attached to a
micromanipulator that works like the spring of a dynamometer: after
measuring the stiffness of the spring, forces are deduced from
extension measurements. Examples of micromanipulators include atomic
force microscopy cantilevers (Moy et al., 1994, Carrion-Vazquez et al, 1999), glass fibers
(Kishino and Yanagida, 1988; Cluzel et al.,
1996), biomembrane force probes (Evans and Ritchie,
1997; Merkel et al., 1999), and microbeads held
by optical tweezers (Block et al., 1990; Finer et
al., 1994; Wuite et al., 2000a). Typical
stiffness ranges from 1 N/m for the former to 105 N/m for
the latter. For measuring biological forces, the typical extensions
that must be detected are consequently of a few nanometers; a distance
also characteristic of the step-size of molecular motors (Schnitzer and Block, 1997). In this article, we report
a new kind of micromanipulator in which micrometric particles are
monitored and manipulated in three dimensions (3D) using magnetic field gradients and servo loops. Our apparatus fulfills all the single molecule biophysics requirements previously mentioned and presents an
alternative to the various existing dynanometers and manipulators.
The early setups able to manipulate magnetic objects in solution were
constructed by biophysicists for the in vivo study of the viscoelastic
properties of the cytoplasm (Crick and Hughes, 1949;
Yagi, 1960). More recently, this technique has been
applied to the rheology of actin filament solutions. After the first
experiments by Sackmann and co-workers (Ziemann et al.,
1994), in which the motion of magnetic particles was confined
to a single horizontal axis, Amblard et al. (1996a,b)
built a micromanipulator for precise and easily controlled
two-dimensional translation and rotation of micrometric beads.
Independently, magnetic piconewton-force transducers have been used to
investigate the elastic behavior of phospholipidic membranes
(Hein-rich and Waugh, 1996; Simson et al.,
1998). Forces ranging from hundreds of femtonewtons to nanonewtons were measured, but micromanipulation of the particle was
not possible. A somewhat similar apparatus was recently described (Haber and Witz, 2000) with a special care to obtain a
uniform force on a large spatial domain (1.5 cm). Very accurate
positioning and force measurements have also been demonstrated using a
macroscopic magnetic particle levitated by a single coil
(Gauthier-Manuel and Garnier, 1997). Pursuing those
works, we report here the design of a mag-netic micromanipulator that
could also be used as a new tool for scientific exploration at the
single-biomolecule level.
In its application, our apparatus is very similar to optical
tweezers (Svoboda and Block, 1994; Simmons et
al., 1996), it allows displacement of small beads (a few
microns in diameter) in solution and to use them as handles or
picodynamometers. The positioning of the particle in 3D is achieved
with a precision of a few nanometers and forces from a few tenths to
tens of piconewtons are simultaneously measured. In the optical
tweezers experiment, a particle having a larger refractive index than
its surrounding medium is trapped by the radiation pressure of a
focused laser beam. The intensity profile of the beam corresponds to a
real potential well that traps the bead in a precise location. In our setup, a system of electromagnets creates field gradients, producing a
force on a super-paramagnetic object. Adjusting the current running
through the coils allows us to change the intensity and the direction
of this force. Furthermore, by combining, in a feedback loop, this
manipulator with a video-positioning system, we are able to control the
3D position of the particle in real time. Note that, in this case, the
action of the magnets is global and the particle is, in fact, trapped
in a virtual potential well by the servo loop. Additionally, we may
rotate the object while holding it fixed because the direction of the
field imposes the angular orientation of the particle magnetic dipole.
Finally, after calibration of the apparatus, the force acting on the
particle can be determined by measuring the currents driving the electromagnets.
The magnetic tweezers described here share most of the features
of optical tweezers while offering the advantage of angular positioning. Moreover, this apparatus does not require a laser that
might photodamage the biomaterial (Liu et al., 1996;
Neuman et al., 1999). We believe that it could, in the
future, meet cell biologists' needs and allow them precise positioning
of organelles, in vivo microrheological investigations (Crick
and Hughes, 1949; Yagi, 1960), and force
measurements (Wang et al., 1993; Guilford et al.,
1995).
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DESIGN OF THE APPARATUS |
Principle
The apparatus consists of two distinct parts: a 3D positioning
algorithm (discussed later in this section) and a set of electromagnets allowing the 3D displacement of the studied particle (discussed in the
next subsection).
As described in Fig. 1, the cell
containing the magnetic particles in solution is held on the stage of
an inverted microscope. This cell is typically a small capillary tube
with a rectangular section (thickness of 300 µm; Vitrocom, Mountains
Lakes, NJ). Its top and bottom surfaces are of good optical quality. A
system of six vertical electromagnets with their pole pieces arranged in a hexagonal pattern is placed just above the capillary tube. Parallel light illuminates the sample through a 2-mm-diameter aperture
located at the center of the hexagon. An xyz translation stage allows the accurate positioning of the electromagnets with respect to the optical axis of the objective.
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FIGURE 1
General magnetic tweezers setup. A thin sample is
observed with an inverted microscope, a CCD image is processed by a
computer that drives the electromagnets to servo the bead position in
real time.
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During micromanipulation, a magnetic particle is located with nanometer
accuracy by video analysis. The computer program determines its
position in the three spatial dimensions at video rate. Then the
digital feedback loop adjusts the current in each electromagnet to
cancel the difference between the desired and the observed positions of
this bead. The six-fold symmetry of the electromagnets allows rotation
of the direction of the magnetic field and hence of the magnetic
particle itself. The force applied to the bead can be directly
evaluated by Brownian motion analysis (Strick et al.,
1996; Allemand, 1997). This method allows the
detection of forces ranging from tens of femtonewtons to tens of
piconewtons. Alternatively, the force can be read from the currents
driving the coils. This requires previous force calibration against the viscous drag or against the Brownian fluctuations.
The present apparatus is inspired from our previous setup using
permanent magnets where position and angle were controled through
simple motorized stages (Strick et al., 1998), allowing stretching and twisting of DNA molecules. The electromagnets allow faster control, which is necessary for operation in a tweezers mode.
Magnetic field
Coils and pole pieces
The six vertical coils (Fig. 2) are
attached to a soft steel base and are capped by curved pole pieces. The
soft steel ring is designed to close the field lines in the system,
thus increasing the magnitude of the magnetic field. A cylinder of
Plexiglas fixed at the pole-pieces end improves the mechanical cohesion
of the apparatus.
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FIGURE 2
Detailed mechanical setup of the electromagnets. The
six coils are placed in a hexagonal geometry, the magnetic field
gradients occur between the tips of the mumetal pole pieces. With this
configuration, the magnitude and direction of the force acting on the
bead can be altered by modulating the driving currents in the coils.
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The round piece closing the field lines is made of XC15 soft steel
(Tonnetot Metaux, Fontenay-sous-Bois, France) and the pole pieces are
cylinders of mumetal (Goodfellow, Cambridge, U.K.). Those two alloys
were chosen for their low remanent magnetization (20 gauss for
mumetal). Coils (Lima 600880, Vizenza, Italy) are made of copper wire
and have a resistance of 10 ± 0.2 
. Each of them is driven by
a current-power amplifier connected to the computer by a
digital-to-analog converter. To avoid magnetic hysteresis, each change
of the coil-driving current is accomplished with an exponentially
decaying oscillating component added (the amplitude being the change
size divided by two at each period, i.e., every 4 ms). With low
hysteresis materials, this method assigns a unique value of the
magnetic field to the driving current. A more sophisticated method has
been used in other experiments (Amblard et al., 1996b), where an active control of the magnetic field through Hall probes fixed
at the end of each pole piece feed back the current in the associated coil.
Current configurations and bead movement
The coils have been designed to produce on the bead a force whose
three components may be adjusted independently. Furthermore, various
experimental constraints had to be overcome: the microscope objective
and the light illumination path did not allow placement of coils along
the vertical optical axis; some symmetry had to insure the rotation
ability. We have found that a set of six vertical electromagnets placed
in hexagonal pattern over the sample was adequate. However, the present
system can only apply a force directed upward; the downward motion of
the bead relies upon gravity. This is certainly a limitation in this
apparatus, but it can be overcome in the future by a more complex set
of electromagnets. The six coils used here represent a minimal system
that, for the sake of simplicity, will be discussed in this paper.
We will first describe how a force is generated in the z
direction and then along the x and y axis. Let us
assume that the six coils are numbered clockwise from 0 to 5 with coils
1 and 4 laying along the y axis. To produce a vertical
force, we generate a horizontal magnetic field collinear to
y and varying strongly along z (Fig.
3). If the three coils 0, 1, and 2 are
run by a current Iz and the three opposite ones
3, 4, and 5 by a current 
Iz (Fig. 4
A), a vertical force will be
applied on every bead located close to the center of the hexagon. This
force can easily overcome the weight of each particle and lift it.
Beads are then simply returned to the ground by reducing the currents:
they fall under the effect of the gravity. A feedback loop adjusting
Iz can finally stabilize the vertical position
of a selected bead. However, the levitated particle will rapidly
diffuse in x and y.
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FIGURE 3
Coils and pole pieces produce a horizontal magnetic
field in the middle of the sample. The magnetic moment  of the
bead aligns with the field lines and the vertical magnetic field
gradient exerts a force  that raises the
super-paramagnetic object.
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FIGURE 4
Schematic representation of the principles governing
the bead displacement (top view of the sample). (A, B, and
C) Three different current configurations are used to move
the magnetic particle along z, x, and y.
(D) More complicated displacements can be reached by linear
combination of the basic settings. Note that, in this figure, all the
currents (Ix, Iy, and
Iz) are positive.
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Now, suppose that the mean value of the current
Iz creates a force that exactly equilibrates the
effects of gravity (Fig. 4 A). If an excess current
Ix runs in coils 2 and 3 while coils 0 and 5 have their driving current reduced by the same amount (Fig. 4
B), the x axis symmetry is broken and a
horizontal field gradient is generated: the bead moves along the
x axis toward the coils producing the strongest fields.
Comparing the Ix and Ix
configurations, it is clear that the vertical force is the same,
whereas the horizontal one changes its sign. If this
Ix is not too large, the horizontal force acting
on the particle is then proportional to Ix
while, for symmetry reason, the vertical component of the magnetic
force has zero linear dependence and is only affected to the second order.
Motion along the y axis may be obtained from the initial
configuration by adding a current Iy in the two
coils 1 and 4, so that Iz + Iy flows in 1 and Iz + Iy in 4. The magnetic field is then reinforced
around coil 1 and reduced around coil 4: as a result the particle moves
toward the strongest electromagnet (Fig. 4 C). Again, the
horizontal component of the force is proportional to
Iy while the vertical one varies only
quadratically with this additional current.
The sensitivity factor between the two axes differs, the symmetry
breaking in x being stronger than in y. Oblique
displacement of the bead (Fig. 4 D) may also be obtained
through a linear combination of the two previous perturbations
Ix and Iy (Table
1)
Despite Iz being only positive, for each bead
one finds a characteristic current I0 above
which the bead rises and below which it falls. Both
Ix and Iy may be positive
or negative, but, as detailed below, their absolute values had been set
proportional to Iz. Finally, it is worth noting
that nearly horizontal force can be applied along x by
altering the current configuration as described in Table 1.
Video acquisition and data treatment
Images of the sample are collected through the 100× oil immersion
objective of an inverted microscope (Leica DMIRBE). A CCD camera (Sony
XC-77CE) operating in 50-Hz field mode sends the data to a video
acquisition card (ICPCI, Imaging Tech., Bedford, MA) installed in a computer.
Three-dimensional tracking of the bead is achieved in real time by a
computer program. Typically, we analyze 25 images per second but twice
this acquisition speed could even be reached by alternately using even
and odd video frames. Evaluation of the particle displacement from one
field to the next is done with sub-pixel resolution for time lapses of
a few seconds (Allemand, 1997). At longer time scales,
the experimental noise has a 1/f component that decreases
the precision. The x, y positions are first obtained by
real-time correlation of the bead images (Gelles et al.,
1988). Then the z position is obtained by using
parallel illumination: the bead image is surrounded with diffraction
rings the diameter of which increases with the distance of the particle from the focal plane. The x, y position is measured with an
accuracy of a few nanometers whereas z is determined with a
10-nm resolution (see Appendix for more details).
Digital feedback loop
Digital proportional-integral feedback loops are used to lock a
particle in a given position. In the horizontal plane, the currents
Ix and Iy are chosen to
be proportional to the main current Iz and are
calculated as
|
(1)
|
In this equation, u corresponds to the error signal
between the present position of the particle and the set one;
Pu and Ku are,
respectively, the proportional and integral coefficients, 
(u) is the sum over the previous error signals; and
Cu is the normalized correction signal.
Using only a proportional correction is equivalent to generating a
force proportional to u, i.e., attaching the bead to a virtual spring whose stiffness ku is directly
determined by Pu. Writing
Au, the proportionality factor between the force
and the driving current associated with direction u,
Fu = Au · Iu, we have
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(2)
|
Adding an integral term (Ku (u)) is important to stabilize the bead to its exact
reference position in the presence of a constant and continuously
applied force (e.g., gravity). In this case, Eq. 2 becomes
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(3)
|
The feedback in the z direction is done by monitoring
Iz. However, the force applied on the bead is
not a linear function of this current. In the low force regime
(Fz < 1 pN), the magnetic field does not
saturate the bead magnetization and thus Fz
varies like I2z
(Fz = AzI2z; see
below, next section). To insure a correct feedback, we then apply a
square-root function to the error signal,
|
(4)
|
with I0 being the current just required to
equilibrate the bead weight.
When the forces applied to the bead are small, the previous relations
can be linearized around their means values,
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(5)
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Thus, the tweezers vertical stiffness is given by
|
(6)
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CHARACTERIZATION OF THE PASSIVE TWEEZERS |
Before describing the tweezers mode, let us discuss the forces
generated by the electromagnets. They may be conveniently characterized by suppressing the feedback loop while maintaining the bead in position
by tethering it with a DNA molecule. Indeed, this method allows
application of strong forces in any direction.
We have prepared both 
and pXII DNA molecules (resp. ~16 and 5 µm long) with one extremity labeled with digoxigenin and the other
with biotin. Incubating these molecules with streptavidin-coated super-paramagnetic beads 4.5 µm in diameter (Dynabeads, Dynal, Oslo,
Norway) results in the attachment of the particle to the biotin end of
the biopolymer. Injection of these beads into an antidigoxigenin-coated
glass capillary allows the dioxigenin end of the DNA to attach to the
tube surface.
Force measurements along z
We have first used the electromagnets configuration, which
produces a force only along the z axis (Fig. 4
A). In this situation, the bead behaves as an inverted
pendulum immersed in a thermal bath at temperature T. As we
have shown in our previous work (Strick et al., 1996),
the analysis of the horizontal Brownian motion of the particle permits
measurement of the stretching force. More precisely, the
bead-positioning software determines the DNA extension l and
the particle transverse fluctuations x. Using the
equipartition theorem, the vertical magnetic force
Fmag may then be evaluated through the simple
formula, Fmag = kBTl/
x2
(Fig.
5).
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FIGURE 5
Principle of force measurement. The vertical magnetic
force Fmag applied to the bead stretches the DNA
molecule. The transverse Brownian fluctuations
x2 of this inverted pendulum are then
used to evaluate its rigidity Fmag/l and thus
the pulling force Fmag.
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By ramping the current in the coils, we could construct the force
versus extension curve of the DNA molecule (Fig. 6
A). The tweezers develop a
force along z varying from 50 fN to 20 pN. Below 1 pN, we
found the expected Fz I2z behavior
characteristic of unsaturated magnetic materials (Fig. 6 B);
then, around 20 pN, saturation occured. This maximum force is only five
times smaller than the one we were able to apply using permanent
magnets (Allemand et al., 1998). Higher forces could
certainly be produced by using magnetic materials with higher
saturation field.
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FIGURE 6
(A) Force versus extension curve measured
for a 4.5-µm bead attached to a DNA molecule. The full line is a
fit to the worm-like chain model with a persistence length of 50 nm.
(B) The current Iz producing the
corresponding vertical force Fz. Data were
collected with different DNA molecules.
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Modulation of the force direction
Taking advantage of the DNA nonlinear elasticity described by the
worm-like chain model (Bouchiat et al., 1999), we
investigated the ability of the tweezers to pull particles in an
arbitrary direction. Indeed, for stretching forces larger than 1 pN,
the length of the dsDNA molecule varies little and the position of the
bead relative to the biopolymer-anchoring point indicates the force direction.
As may be seen in Fig. 7, the
stretching-force direction sweeps a very large angle in the
x direction while keeping a modulus 
5 pN as determined by
Brownian motion analysis. The pulling angle reaches ±70° at high
stretching force where the weight of the bead is negligible
(F > 1 pN). In the y direction, similar
results are obtained, but the pulling angle only reaches ±50°. At
lower forces, the weight of the bead combines with the magnetic force and allows the resulting force to point in all spatial directions. Pulling a DNA molecule nearly horizontally could be useful, for instance, to visualize enzymes moving along this polymer.
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FIGURE 7
Positions of the bead center for the full range of
x current modulation. The particle, 4.5 µm in diameter, is
tethered to the glass surface by a pXII DNA molecule ~5 µm long.
The position of the bead and of the DNA molecule are drawn in vertical
position and for the maximum modulations. The circle is a fit to the
data points obtained for moderate modulation (+). These points are
typically within 20 nm away from the circle, demonstrating that the
pulling force is unchanged. In the extreme modulations ( ) the
pulling direction is nearly horizontal and is limited by the fact
that the bead touches the glass surface. For those extreme modulations,
the stretching force decreases slightly as indicated by the shorter
extension. The two long dashed lines indicate the boundary between the
normal and the altered configuration of the coil-driving currents (see
Table 1).
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CHARACTERIZATION OF THE ACTIVE TWEEZERS |
Ideal tweezers should allow manipulation of a micrometric bead
while giving access to the three force components at the same time.
This is nearly achieved in the present setup by recording the
electromagnets driving currents. Nevertheless, it is of course necessary to previously perform tweezers calibration, i.e., to investigate the relation between currents and forces. In the
small-force regime, these relations are linear, and thus the
calibration consists of measuring the different proportionality
constants Au. These coefficients vary from one
magnetic bead to another. However, we will show below that such a
calibration can be achieved easily by recording the particle
fluctuations in the trapped state with no external force applied.
To explain the tweezers properties and the related calibration
procedure, we first introduce a simplified model with only an
instantaneous proportional feedback. This model depends only on two
parameters: the tweezers elastic stiffness ku
and the particle viscous drag coefficient r. We show
that the analysis of the bead fluctuations in Fourier space allows
determination of r at high frequencies and
ku at low frequencies. The measurement of
r leads to the viscosity of the fluid , and the
measurement of ku leads to the
Au through Eq. 2. Consequently, the tweezers may
be used in two different operating modes: as a viscosimeter or as a dynamometer.
Finally, we discuss the properties of the real apparatus with its
slower digital feedback and its integral correction. In this case, the
tweezers cannot be described anymore by our simplified model, but are
better characterized by a recursive equation accounting for the servo
loop delay. Within this new context, we present three complementary
calibration methods that provide absolute measurements of the force.
Because the techniques and models used here are sometimes similar to
the ones used for optical tweezers calibration, the following analysis
can be read in parallel with the reviews, Svoboda and Block
(1994), and Gittes and Schmidt (1998).
Model for ideal tweezers
To study the complete feedback loop of our apparatus, a 4.5-µm
super-paramagnetic bead is locked 10 µm above the surface. As seen
above, couplings among the x, y, and z forces are
only quadratic, and, consequently, the three trapping directions may be
considered as independent. For the sake of simplicity, we will also
first work with a proportional feedback. Furthermore, we will assume
that the feedback presents no delay, which allows a simple description
of "ideal" tweezers. The bead is locked in a virtual
one-dimensional potential well where the magnetic tweezers respond with
a force Fu = ku · u to a deviation u from the
initial set position. The equation of motion of the particle can
thereby be written,
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(7)
|
where m is the mass of the bead, r its
radius, the viscosity of the solution, and the viscous drag
coefficient (6 for a spherical object far from any surface). It is
easy to verify that the system is overdamped and the inertial term may
thus be omitted. FL is the stochastic Langevin
force responsible for the fluctuations characteristic of the Brownian
motion of the particle. From the fluctuation dissipation theorem, it
follows that FL(t) = 0 and
FL(t)FL(t') = 2kBTr(t t'), which appears as a white noise in Fourier space,
|FL(f)|2 = 4kBTr. Note that this noise is
also the intrinsic noise of our measurement.
In frequency space, the density of fluctuations is then given by
|
(8)
|
where fc = ku/(2r). This power spectrum
is a Lorentzian corresponding to the response function of a bead
attached to a spring, immersed in a viscous medium, and excited by a
white noise (the Langevin force). This noise depends only on
dissipative terms, it is proportional to and r. More
precisely, at low frequencies (f 
fc), the power spectrum presents an asymptotic
white noise determined by the excitation of the spring
ku through the Langevin noise, whereas, at high
frequencies (f 
fc), the
f2 behavior is dominated by the viscous
term r (Fig. 8 at small Pz values).
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FIGURE 8
Data points. Average power spectra of the
vertical position fluctuations of a trapped bead (only a variable
proportional feedback is applied here). These power spectra have a
Lorentzian shape when the feedback is small (top curves). As
the feedback is increased, the fluctuations decrease and the cutoff
frequency increases. At high feedback, ringing occurs (bottom
curves). Solid lines. Power spectra obtained from the
iterative model. L, , and were
first determined by fitting the lowest power spectrum to Eq. 16 ( can only be evaluated when ringing occurs, i.e. at high
Pz). The other spectra were then fitted while
keeping L and equal to the found
values. Finally, the proportionality between and
Pz ( = AzI02Pzt/r
[Eqs. 6 and 13]) was checked.
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Viscosimeter mode
Because, at high frequencies, the Brownian fluctuations of the
bead presents a 1/f2 regime (see Fig. 8),
the spectrum of the velocity fluctuations (the derivative
Vu) presents an asymptotic white noise, the
value of which is proportional to the object diffusion coefficient
D and thus inversely proportional to the viscous term
r,
|
(9)
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Consequently, the particle Brownian motion offers a simple means
to use the magnetic tweezers as a viscosimeter in vitro (Ziemann
et al., 1994, Amblard et al., 1996b) or in vivo
(Crick and Hughes, 1949; Sato et al.,
1984; Zaner and Valberg, 1989, Yagi,
1960). First, the tweezers bring a bead to a specific point of
interest. Then, the feedback is switched to a low-stiffness mode, which
allows measurement of the local viscosity while keeping the probe in a
defined area. Accurate data may be obtained at such low-feedback
parameters because the cutoff frequency is small, leaving a wide
white-noise regime in the bead-velocity spectrum. Additionally, care
must be taken to compensate for the video camera filtering due to
exposure-integration damping of the frequencies close to the
acquisition rate.
Tweezers mode
The measure of ku leads to the
calibration of the apparatus as a dynamometer. When the tweezers can be
considered as ideal, the calibration procedure can be achieved by using
the equipartition theorem. Provided that we record the bead
fluctuations with an infinite frequency range, we can write
kBT/2 = kuu2/2 in real
space. Thus, in Fourier space, we have
|
(10)
|
This calibration method is very powerful, but, as explained below,
it applies to the magnetic tweezers only when the values of the
feedback parameters are low (i.e., when the Pu
are small). We will first discuss here the intrinsic limitations of the
method and its validity conditions. Then, we will show that, although this method cannot be used directly when the values of the feedback parameters are high, it may be adapted to this situation.
The two intrinsic limitations of the equipartition calibration are
finite bandwith and accuracy. In all experiments, the signal bandwidth
is limited at high frequency to the half of the sampling rate
fs/2 and at low frequency to the inverse of the
observation window fL = T1. Thus, the integration limits in Eq. 10
becomes fL and fs/2
instead of 0 and 
. To maintain the accuracy of this relation, we
must either ensure that fs/2
fc fL or correct
the equation from the limited integration range. This last method
requires evaluation of the integral of the Lorentzian within the
experimental range to determine fc.
Independently, it is worth noting that the estimation of
fc is also useful to evaluate the statistical
error on u2 and thus on
ku. To obtain good statistics and a precise
measurement of the trap stiffness, Brownian motion must be recorded
long enough. ku is given with an accuracy of
1/N if the fluctuations are analyzed over a period
N2 times larger than
1/fc. In practice, we always adjust the
measurement time for reaching errors lower than 10%, for example, a
stiffness of 107 N/m
(1/(2fc) = 0.42 s) is evaluated with a
16,384-frames acquisition at 25 Hz, (total time T = 655 s), leading to an accuracy of
.
In our experiment, as may be seen in Fig. 8, the spectra are indeed
Lorentzian when the proportional feedback coefficient Pu is not too large. In this regime, the
simplified model of the tweezers may be applied, and the measure of
ku using the equipartition theorem is valid (see
Fig. 9). Nevertheless, as
Pu is increased, this method does not remain
valid anymore, and we can only use the low frequency asymptotic part of
the spectrum to evaluate ku. In fact, it appears
clearly that the value of the low-frequency white noise scales with
P
, whereas the cutoff frequency
fc increases linearly. As a consequence, when
Pu increases, the trap characteristic time
1/(2fc) decreases, and, above a critical
value Puc, it becomes significant compared with
the delay in digital feedback (roughly two video frames). Resonance
then occurs, leading to an instability that forbids higher feedback
parameters and restrains the use of Eq. 10 at small stiffness values
(maximal trap stiffness of ~107 N/m).
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FIGURE 9
Mean square fluctuations versus the tweezers stiffness
kz (top axis) and the proportional
feedback coefficient (Eq. 13, bottom axis). For ideal
tweezers, the equipartition theorem applied to the simple model (Eqs. 8
and 10) gives a straight line (dashed line). For real
tweezers, the mean square fluctuations calculated from the experimental
spectra of Fig. 8 (squares) coincide with the prediction of
the digital feedback model (solid line, calculated from Eq. 16 with L and as determined in Fig.
8). As the feedback proportional coefficient (i.e., the stiffness) is
increased, the fluctuations decrease until they start to increase again
because ringing occurs in the servo loop.
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Effect of the digital feedback loop delay
To correctly describe the servo loop at high-feedback parameters,
we must work with discrete times and recurrent relations. Let
un be the position of the bead at image
n, this parameter is driven by the equation,
|
(11)
|
where Ln is the displacement due to the
Langevin force. As we can see, the bead position un+1
at image n + 1 depends on its position at image
n but also on its position at image n through the feedback correction un 
. Indeed, due to the delay between the video acquisition and the control of the magnetic field, the harmonic correction to the position
at time n could only proceed using a previous particle location. Typically, the video camera has a delay of one image, and the
acquisition hardware and software are responsible for the delay of an
additional image, leading to ~ 2. Finally, during the time
lapse t of one image (40 ms here), the spring force kuun is balanced by
the sole viscous drag, and, therefore, the equation of the dynamic
leads to
|
(12)
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i.e.
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(13)
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It is convenient to consider Eq. 11 in Fourier space where the
Brownian fluctuations of the Langevin force are
|
(14)
|
Note that L20 is a noise
density expressed in µm2Hz1. The bead
displacement may then be written,
|
(15)
|
with an amplitude given by
|
(16)
|
This equation would be valid for a camera sampling the image for
an infinitely small amount of time at each acquisition. Standard
cameras integrate light over a significant fraction
ti of the acquisition period t.
This averaging, together with the aliasing, introduces a filtering that
slightly damps the signal at high frequency (Gittes and Schmidt,
1998). Consequently, we are obliged to introduce a correction
Cr(f) in each data processing (see Appendix.).
Fitting Eq. 16 multiplied by Cr(f) to the experimental data,
we obtain the values of the three parameters
L20, , and . The fit is quite
good and describes correctly the ringing behavior occurring when the
proportional feedback Pu is large (Fig. 8). More
precisely, at a given , it is the parameter that determines the
system behavior: if it is too high, i.e., if fc
reaches one tenth of the sampling frequency (Eq. 13), the bead starts
to oscillate (Fig. 9).
In summary, the stronger the feedback the higher the effective spring
constant ku is and the faster the bead reacts.
However, the delay due to the video acquisition and treatment limits
this feedback. Typically, the maximum frequency at which the feedback loop can safely operate is 
of the acquisition frequency.
Effect of the integral correction
The integral component
Ku 
(u) of the feedback loop
allows, for instance, delivery of a constant current
Iz, which levitates the bead even though the
error signal in z is zero. Introducing the integral feedback
requires modification of Eq. 11 as follows:
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(17)
|
with = Ku/Pu. In Fourier space,
this leads to
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(18)
|
Again, this is the response using an ideal camera. With a real
camera, we must correct this equation by Cr(f) (see Eq. 32). The coefficient must be small because it introduces a phase shift.
Proper values should be kept smaller than a tenth. In these conditions,
the lowest frequencies of the bead fluctuations are strongly filtered
out (see Fig. 12 A) and the stability of the system is improved.
When the integral feedback is used, the stiffness of the tweezers
ku depends on the frequency, and is thus not
rigorously defined. However, we will still use
ku to characterize the stiffness; this effective
ku being the one measured if is set to zero
while all other parameters are kept constant.
Calibration against the viscous drag
The numerical results obtained by the different calibration
methods are discussed in Comparison of the Three Calibration Methods below. Viscous drag is often used to calibrate optical tweezers. In our
case, it does not require mechanically moving any part of the setup. We
simply rely on the software to change the virtual potential well
position. A 4.5-µm super-paramagnetic bead is locked in x,
y, and z at 10 µm above the surface. We then move the
set particle position in u by using a square wave signal,
while we limit the maximum electromagnet current
Iu to an absolute value Ium (Fig. 10
A). This threshold thereby
defines the maximum bead velocity Vu in
direction u. The object trajectory may be seen in Fig. 10
B, it is a slew rate-limited square wave that is repeated several times to allow averaging and then noise reduction. By varying
Ium, we construct the calibration curve that
relates the bead velocity to the driving current. As seen in Fig.
11, Vu and Iu are proportional, and we can thus define the
measured proportionality constant,
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(19)
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Because the force Fu acting on the
bead is linked to its velocity by the Stokes' law,
Fu = r · Vu, it becomes
proportional to the driving current Iu and, from
Eq. 2, we have
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(20)
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The quantity r may be obtained either by
optically measuring the bead radius and knowing the fluid viscosity or,
in an absolute determination, by using the high-frequency bead
fluctuations as described previously.
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FIGURE 10
Procedure for tweezers calibration using Stokes' law.
(A) Averaged coil-driving current producing the back and
forth bead motion. (B) Corresponding averaged oscillations
of the bead with constant moving velocity. The dash line corresponds to
the imposed well position. When the calibration is achieved, the
driving current may be converted in force as done on the right axis of
panel A.
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FIGURE 11
Calibration along y using Stokes' law. We
have reported the bead velocity Vy versus
Iym, the maximum value allowed on the driving
signal. Note that the linear behavior is even found for large
modulation. Small driving currents are inaccessible because they
prohibit normal operation of the feedback loop.
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The calibration along z can be done by driving
Iz. However, because the range of measurement in
the vertical direction does not exceed 10 µm (i.e., the size of the
calibration image), special care must be taken to avoid losing the bead.
Calibration using the Brownian fluctuations
When working with micrometric objects in fluids, Brownian
fluctuations are important. Although they usually limit the accuracy of
any measurement, they offer here two alternative calibration methods
that are complementary to the previous one.
The calibration procedure is completely achieved during particle
locking by the magnetic tweezers. First, the computer records, simultaneously, the spontaneous fluctuations of the bead
u(t) and the associated driving currents
Iu(t). In the absence of external perturbations,
the trapped particle is stable in position because the feedback exactly
compensates the Langevin random force at frequencies lower than the
system characteristic cutoff frequency fc (Fig.
12).
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FIGURE 12
Calibration along y using the
Brownian fluctuations of a trapped bead. (A) Experimental
power spectrum of the y component of the bead fluctuations
(dots) and fit to the feedback response defined by Eq. 18.
(B) Power spectrum of the bead velocity fluctuations
Vy. The high frequencies plateau at 0.395 µm2Hz1 is related to r via
Eq. 9. (C) Power spectrum of the coil-driving current
Iy in the same conditions (dots) and
predicted current using the digital feedback model (solid
line, calculated from Eqs. 1 and 18). The low-frequency asymptotic
regime corresponds to the current required to compensate for the
Langevin force.
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Asymptotic spectrum analysis
At low frequency, the feedback reaction equilibrates the Langevins
force, F
(f
fc) = F
. Thus we
have, in x or y,
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(21)
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In z, for the linearized regime, where Eq. 5 holds, we
have
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(22)
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As seen in Eq. 9, the high frequency bead velocity fluctuations
V(f fc)
can be used to calculate r. For u = x or y, we then obtain
|
(23)
|
and, using Eq. 20,
|
(24)
|
Along the z direction, we have
|
(25)
|
and
|
(26)
|
Complete fit of the digital feedback response
As shown in Figs. 8 and 12, the entire power spectrum of the bead
fluctuations may be fitted to the digital feedback response described
by Eq. 18. This provides the values of the three relevant parameters
L
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ABSTRACT
LIST OF SYMBOLS
