When a dielectric object is placed between two opposed,
nonfocused laser beams, the total force acting on the object is zero but the surface forces are additive, thus leading to a stretching of
the object along the axis of the beams. Using this principle, we have
constructed a device, called an optical stretcher, that can be used to
measure the viscoelastic properties of dielectric materials, including
biologic materials such as cells, with the sensitivity necessary to
distinguish even between different individual cytoskeletal phenotypes.
We have successfully used the optical stretcher to deform human
erythrocytes and mouse fibroblasts. In the optical stretcher, no
focusing is required, thus radiation damage is minimized and the
surface forces are not limited by the light power. The magnitude of the
deforming forces in the optical stretcher thus bridges the gap between
optical tweezers and atomic force microscopy for the study of biologic materials.
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INTRODUCTION |
For almost three decades, laser traps have been
used to manipulate objects ranging in size from atoms to cells (Ashkin,
1970
; Chu, 1991; Svoboda and Block, 1994). The basic principle of laser traps is that momentum is transferred from the light to the object, which in turn, by Newton's second law, exerts a force on the object. Thus far, these optical forces have solely been used to trap an object.
The most common laser trap is a one-beam gradient trap, called optical
tweezers (Ashkin et al., 1986). Optical tweezers have been an
invaluable tool in cell biological research: for trapping cells (Ashkin
et al., 1987; Ashkin and Dziedzic, 1987), measuring forces exerted by
molecular motors such as myosin or kinesin (Block et al., 1990;
Shepherd et al., 1990; Kuo and Sheetz, 1993; Simmons et al., 1993;
Svoboda et al., 1993), or the swimming forces of sperm (Tadir et al.,
1990; Colon et al., 1992), and for studying the polymeric properties of
single DNA strands (Chu, 1991).
In contrast, the optical stretcher is based on a double-beam trap
(Ashkin, 1970; Constable et al., 1993) in which two opposed, slightly
divergent, and identical laser beams with Gaussian intensity profile
trap an object in the middle. This trapping is stable if the total
force on the object is zero and restoring. This condition is fulfilled
if the refractive index of the object is larger than the refractive
index of the surrounding medium and if the beam sizes are larger than
the size of the trapped object. In extended objects such as cells, the
momentum transfer primarily occurs at the surface. The total force
acting on the center of gravity is zero because the two-beam trap
geometry is symmetric and all the resulting surface forces cancel.
Nevertheless, if the object is sufficiently elastic, the surface forces
stretch the object along the beam axis (see Fig.
1) (Guck et al., 2000). At first, this
optical stretching may seem counterintuitive, but it can be explained
in a simple way. It is well known that light carries momentum. Whenever
a ray of light is reflected or refracted at an interface between media
with different refractive indices, changing direction or velocity, its
momentum changes. Because momentum is conserved, some momentum is
transferred from the light to the interface and, by Newton's second
law, a force is exerted on the interface.
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FIGURE 1
Schematic of the stretching of a cell trapped in the
optical stretcher. The cell is stably trapped in the middle by the
optical forces from the two laser beams. Depending on the elastic
strength of the cell, at a certain light power the cell is stretched
out along the laser beam axis. The drawing is not to scale; the
diameter of the optical fibers is 125 ± 5 µm.
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To illustrate, let us consider a ray of light passing through a cube of
optically denser material (see Fig. 2).
As it enters the dielectric object, the light gains momentum so that
the surface gains momentum in the opposite (backward) direction.
Similarly, the light loses momentum upon leaving the dielectric object
so that the opposite surface gains momentum in the direction of the light propagation. The reflection of light on either surface also leads
to momentum transfer on both surfaces in the direction of light
propagation. This contribution to the surface forces is smaller than
the contribution that stems from the increase of the light's momentum
inside the cube. The two resulting surface forces on front and backside
are opposite and tend to stretch the object (Guck et al., 2000).
However, the asymmetry between the surface forces leads to a total
force that acts on the center of the cube. If there is a second,
identical ray of light that passes through the cube from the opposite
side, there is no total force on the cube, but the forces on the
surface generated by the two rays are additive. In contrast to
asymmetric trapping geometries, where the total force is the trapping
force used in optical traps, the optical stretcher exploits surface
forces to stretch objects. Light powers as high as 800 mW in each beam
can be used, which lead to surface forces up to hundreds of
pico-Newton. There is no problem with radiation damage to the cells
examined, which is not surprising because the laser beams in the
optical stretcher are not focused, minimizing the light flux through
the cells in comparison to other optical traps (see Viability of
Stretched Cells). To demonstrate this concept of optical deformability, we stretched osmotically swollen erythrocytes and BALB 3T3 fibroblasts.
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FIGURE 2
Momentum transfer and resulting forces on a dielectric
box due to one laser beam incident from the left. (A) A
small portion of the incident light is reflected at the front surface.
The rest enters the box and gains momentum due to the higher refractive
index inside. On the back, the same fraction is reflected and the
exiting light loses momentum. The lower arrows indicate the momentum
transferred to the surface. (B) The resulting forces for
a light power of 800 mW at the front and the back are
Ffront = 105-306 pN and
Fback = 108-333 pN, respectively,
depending on the refractive index of the material. Note that the force
on the back is larger than the force on the front. (C)
Due to the difference between forces on front and back, there is a
total force, Ftotal = Fback  Ffront = 3 27 pN, acting
on the center of gravity of the box. This total force pushes the box
away from the light source. An elastic material will be deformed by the
forces acting on the surface, which are an order of magnitude larger
than the total force.
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Human erythrocytes, i.e., red blood cells (RBCs), were used as initial
test objects. Red blood cells offer several advantages as a model
system for this type of experiment in that they lack any internal
organelles, are homogeneously filled with hemoglobin, and can be
osmotically swollen to a spherical shape. They are thus close to the
model of an isotropic, soft, dielectric sphere without internal
structure that we used for the calculation of the stress profiles (see
below). Furthermore, they are very soft cells and deformations are
easily observed. As an additional advantage, RBCs have been studied
extensively and their elastic properties are well known (Bennett, 1985,
1990; Mohandas and Evans, 1994). The only elastic component of RBCs is
a thin membrane composed of a phospholipid bilayer sandwiched between a
triagonal network of spectrin filaments on the inside and glycocalix
brushes on the outside (Mohandas and Evans, 1994). The ratio between
cell radius 
and membrane thickness h,
/h 
100. This means that the bending
energy is negligibly small compared to the stretching energy (see
Deformation of Thin Shells). Thus, linear membrane theory can be used
to predict the deformations of RBCs subjected to the surface stresses
in the optical stretcher. By comparing the deformations observed in the
optical stretcher with the deformations expected, we quantitatively
verified the forces predicted from our calculations.
The BALB 3T3 fibroblasts under investigation are an example of typical
eukaryotic cells that, in contrast to RBCs, have an extensive
three-dimensional (3D) network of protein filaments throughout the
cytoplasm as the main elastic component (Lodish et al., 1995). In this
network, called the cytoskeleton, semiflexible actin filaments,
rod-like microtubules, and flexible intermediate filaments are arranged
into an extensive, 3D compound material with the help of accessory
proteins (Adelman et al., 1968; Pollard, 1984; Elson, 1988; Janmey,
1991). Classical concepts in polymer physics fail to explain how these
filaments provide mechanical stability to cells (MacKintosh et al.,
1995), but, in most cells, cytoskeletal actin is certainly a main
determinant of mechanical strength and stability (Stossel, 1984, Janmey
et al., 1986; Sato et al., 1987; Elson, 1988).
The actin cortex is a thick (/h 10)
homogeneous layer just beneath the plasma membrane. In cells adhered to
the substrate, additional bundles of individual actin filaments, called
stress fibers, insert into focal adhesion plaques and span the entire cell interior. Dynamic remodeling of this network of F-actin
facilitates such important cell functions as motility and the
cytoplasmic cleavage as the last step of mitosis (Pollard, 1986;
Carlier, 1998; Stossel et al., 1999). Cells are drastically softened by actin-disrupting cytochalasins (Petersen et al., 1982; Pasternak and
Elson, 1985) and gelsolin (Cooper et al., 1987), indicating the
importance of actin. More recently, frequency-dependent atomic force
microscopy (AFM)-based microrheology showed that fibroblasts exhibit
the same viscoelastic signature as homogeneous actin networks in vitro
(Mahaffy et al., 2000). Another experiment (Heidemann et al., 1999)
investigated the response of rat embryo fibroblasts to mechanical
deformation by glass needles. Actin and microtubules were tagged with
green fluorescent protein and the role of these two cytoskeletal
components in determining cell shape during deformation was directly
visualized. Again, actin was found to be almost exclusively responsible
for the cell's elastic response, whereas microtubules clearly showed
fluid-like behavior.
In nonmitotic cells, microtubules radiate outward from the
microtubule-organizing center just outside the cell nucleus (Lodish et
al., 1995). They serve as tracks for the motor proteins dynein and
kinesin to transport vesicles through the cell. Microtubules are also
required for the separation of chromosomes during mitosis (Mitchison et
al., 1986; Mitchison, 1992). Intermediate filaments are unique to
multicellular organisms and comprise an entire class of flexible
polymers that are specific to certain differentiated cell types
(Herrmann and Aebi, 1998; Janmey et al., 1998). For example, vimentin
is expressed in mesenchymal cells (e.g., fibroblasts). Vimentin fibers
terminate at the nuclear membrane and at desmosomes, or adhesion
plaques, on the plasma membrane. Another type of intermediate filament
is lamin, which makes up the nuclear lamina, a polymer cortex
underlying the nuclear membrane (Aebi et al., 1986). Intermediate filaments are often colocalized with microtubules, suggesting a close
association between the two filament networks. Both microtubules and
intermediate filaments are thought to be less important for the elastic
strength and structural response of cells subjected to external stress
(Petersen et al., 1982; Pasternak and Elson, 1985; Heidemann et
al., 1999; Rotsch and Radmacher, 2000). However, intermediate filaments
become more important at large deformations that cannot be achieved
with deforming stresses of several Pascal. Intermediate filaments are
also more important to elasticity in adhered cells as opposed to
suspended cells, where the initially fully extended filaments become
slack (Janmey et al., 1991; Wang and Stamenovic, 2000). Despite these
experiments, a quantitative description of the cytoskeletal
contribution to a cell's viscoelasticity is still missing. The optical
stretcher can be used to measure the viscoelastic properties of the
entire cytoskeleton and to shed new light on the problem of cellular elasticity.
The ability to withstand deforming stresses is crucial for cells and
has motivated the development of several techniques to investigate cell
elasticity. Atomic force microscopy (Radmacher et al., 1996),
manipulation with micro-needles (Felder and Elson, 1990), microplate
manipulation (Thoumine and Ott, 1997), and cell poking (Dailey et al.,
1984) are not able to detect small variations in cell elasticity
because these detection devices have a very high spring constant
compared to the elastic modulus of the material probed. The AFM
technique has recently been improved for cell elasticity measurements
by attaching micron-sized beads to the scanning tip to reduce the
pressure applied to the cell (Mahaffy et al., 2000). Micropipette
aspiration of cell segments (Discher et al., 1994) and displacement of
surface-attached microspheres (Wang et al., 1993) can provide
inaccurate measurements if the plasma membrane becomes detached from
the cytoskeleton during deformation. In addition, all of these
techniques are very tedious and only probe the elasticity over a
relatively small area of a cell's surface. Whole-cell elasticity can
be indirectly determined by measurements of the compression and shear
moduli of densely packed cell pellets (Elson, 1988; Eichinger et al.,
1996), or by using microarray assays (Carlson et al., 1997). However,
these measurements only represent an average value rather than a true single-cell measurement, and depend on noncytoskeletal forces such as
cell-cell and cell-substrate adhesion. The optical stretcher is a new
tool that not only circumvents most of these problems, but also permits
the handling of large numbers of individual cells by incorporation of
an automated flow chamber, fabricated with modern soft lithography
techniques, that guides cells through the detector.
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MATERIALS AND METHODS |
Erythrocyte preparation
The buffer for the RBCs was derived from Zeman (1989) and Strey
et al. (1995) and consisted of 100 mM NaCl, 20 mM Hepes buffer (pH
7.4), 25 mM glucose, 5 mM KCl, 3 mM CaCl2, 2 mM
MgCl2, 0.1 mM adenine, 0.1 mM inosine, 1%
(volume) antibiotic-antimycotic solution, 0.25-1.5% albumin, and 5 units/ml heparin. All reagents were purchased from Sigma (St. Louis,
MO) unless stated otherwise. Red blood cells were obtained by drawing
~10 µl of blood from the earlobe or fingertip. The blood was
diluted with 4 ml of the buffer. Because the buffer has a
physiological osmolarity (~270 mOsm), the RBCs initially have a flat,
biconcave, disc-like shape. However, the buffer was then diluted to
lower the osmolarity to 130 mOsm, at which point the RBCs swell to
assume a spherical shape. The average radius of the swollen RBCs was
measured to be = 3.13 ± 0.15 µm using phase
contrast microscopy. The error given is the standard deviation (SD) of
55 cells measured. The refractive index of spherical RBCs,
n = 1.378 ± 0.005 (Evans and Fung, 1972), the
refractive index of the final buffer was measured to be
n = 1.334 ± 0.001, both of which were used for
the calculations in the RBC stretching experiments.
Eukaryotic cell preparation
As prototypical eukaryotic cells, BALB 3T3 fibroblasts (CCL-163)
were obtained from American Type Culture Collection (Manassas, VA) and
maintained in Dulbecco's Modified Eagle's Medium with 10% nonfetal
calf serum and 10 mM Hepes at pH 7.4. For cells to be trapped and
stretched in the optical stretcher, they must be in suspension. Because
these are normally adherent cells, single-cell suspensions for each
experiment were obtained by incubating the cells with 0.25%
trypsin-EDTA solution at 37°C for 4 min. After detaching, the
activity of trypsin-EDTA was inhibited by adding fresh culture medium.
This treatment causes the cells to stay suspended as isolated cells for
2-4 h. Once in suspension, the cells assumed a spherical shape. Their
average radius was = 9.2 ± 2.8 µm (SD of 20 cells measured), and their average refractive index, n = 1.370 ± 0.005, was measured using index matching in phase
contrast microscopy (see Fig. 3) (Barer
and Joseph, 1954, 1955a,b). The refractive index of the cell medium,
which was used for the calculations of the fibroblast shooting and
stretching experiments, was measured to be n = 1.335 ± 0.002.
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FIGURE 3
Phase contrast image of a BALB 3T3 fibroblast in an
albumin solution with refractive index n = 1.370 ± 0.005. At the matching point, the contrast between cell
and surrounding is minimal. The lighter parts of the cytoplasm have a
slightly lower refractive index, whereas the darker parts have a
slightly higher refractive index than the bulk of the cell. The cell's
radius is 8.4 µm.
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For the fluorescence studies of the actin cytoskeleton of BALB 3T3
cells in suspension, we used TRITC-phalloidin (Molecular Probes, Inc., Eugene, OR), a phallotoxin that binds selectively to
filamentous actin (F-actin), increasing the fluorescence quantum yield
of the fluorophore rhodamine several-fold over the unbound state (Allen
and Janmey, 1994). This assures that predominantly actin filaments are
detected rather than actin monomers. Before staining, the cells were
spun down and gently resuspended in a 4% formaldehyde solution for 10 min to fix the actin cytoskeleton. They were then washed three times
with PBS, permeabilized with a 0.1% Triton-X 100 solution for 2 min,
and washed three more times. Then the cells were stained with a
1-µg/ml TRITC-phalloidin solution for 10 min, followed by a final
washing step (3× with PBS). Fluorescence images were acquired with an
inverted microscope (Axiovert TV100, Carl Zeiss, Inc., Thornwood, NY)
and deconvolved using a Jansson-van Cittert algorithm with 100 iterations (Zeiss KS400 software).
Silica and polystyrene beads
The silica and polystyrene beads used for the calibration of the
image analysis algorithm and for the shooting experiments were
purchased from Bangs Laboratories, Inc. (Fishers, IN). Their radii were
= 2.50 ± 0.04 µm (SD) and = 2.55 ± 0.04 µm (SD), respectively, as given in the
specifications provided by the manufacturer. Using index matching,
their indices of refraction were measured to be n = 1.430 ± 0.003 for silica beads using mixtures of water and
glycerol, and n = 1.610 ± 0.005 for polystyrene
beads using mixtures of diethyleneglycolbutylether and
-chloronaphthalene. The index of refraction of water,
used for the calculation of the forces on the silica and polystyrene
beads in the shooting experiments, was measured to be n = 1.333 ± 0.001.
Experimental setup
The setup of the experiment (see Fig.
4) is essentially a two-beam fiber trap
(Constable et al., 1993). A tunable, cw Ti-Sapphire laser (3900S,
Spectra Physics Lasers, Inc., Mountain View, CA) with up to 7W of light
power served as light source at a wavelength of 
= 785 nm
(30 GHz bandwidth). An acousto-optic modulator (AOM-802N, IntraAction
Corp., Bellwood, IL) was used to control the beam intensity, i.e., the
surface forces. This can be done with frequencies between
10
2 and 103 Hz, thus
allowing for time-dependent rheological measurements in the frequency
range most relevant for biological samples. The beam was split in two
by a nonpolarizing beam-splitting cube (Newport Corp., Irvine, CA) and
then coupled into single-mode optical fibers (mode field diameter = 5.4 ± 0.2 µm, NA 0.11). The fiber couplers were purchased
from Oz Optics, Ltd. (Carp, ON, Canada) and the single-mode
optical fibers from Newport. The optical fibers not only simplify the
setup of the experiment, they also serve as additional spatial filters
and guarantee a good spatial mode quality (TEM00). The maximum light powers achieved in
this setup were 800 mW in each beam at the object trapped. The power
exiting the fiber was measured before and after each experimental run
to verify the stability of the coupling over the 1-2-h period. All
power values given are measured with a relative error of ±1% (SD).
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FIGURE 4
Setup of the optical stretcher. The intensity of the
laser beam is controlled by the acousto-optic modulator
(AOM), split in two by a beam splitter
(BS), and coupled into optical fibers
(OF) with two fiber couplers (FC). The
inset shows the flow chamber used to align the fiber tips and to stream
a cell suspension through the trapping area. Digital images of the
trapping and optical stretching were recorded by a Macintosh computer
using a CCD camera.
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For trapping and stretching cells in the optical stretcher, the
fibers' alignment is crucial. For the RBCs, a solution similar to the
one described in Constable et al. (1993) was used: a glass capillary
with a diameter between 250 and 400 µm was glued onto a microscope
slide, and the fibers were pressed alongside so that they were colinear
and facing each other (not shown in Fig. 4). Red blood cells were so
light that they sunk very slowly and could be trapped out of a cell
suspension placed on top of the fiber ends. For the BALB 3T3
fibroblasts, we used a flow chamber geometry (see inset of Fig. 4) that
allowed us to stream a suspension of cells directly through the gap
between the optical fibers. After successfully trapping one cell, the
flow was stopped and the cell's elasticity was measured. Then the cell
was released and the flow was started again until the next cell was
trapped. The microscope slide, or the flow chamber, was mounted on an
inverted microscope equipped for phase contrast and fluorescence
microscopy. Phase contrast images of the trapping and stretching were
obtained with a CCD camera (CCD72S, MTI-Dage, Michigan City, IN). The
pixel size for all magnifications used was calibrated with a 100 lines/mm grating, which allowed for absolute distance measurements. The side length of the square image pixels was 118 ± 2 nm for the 40× objective with additional 2.5× magnification lens, used for the
cell-size measurements. To measure larger distances, such as the
distance between the fiber tip and the trapped object, we used a 20×
objective, which resulted in an image pixel size of 611 ± 5 nm/pixel. All stretching experiments were done at room temperature.
Image analysis
After completing the experiments, image data were analyzed on a
Macintosh computer to quantify the deformation of a cell in the optical
stretcher. The algorithm was developed in the scientific programming
environment MATLAB (MathWorks, Inc., Natick, MA), which treats bitmap
images as matrices. Figure
5 A shows a typical phase
contrast image of an RBC stretched at moderate light powers (P 100 mW). The boundary of the cell in the image
is the border between the dark cell and the bright halo. The goal was
to extract the shape of the cell from this image to use the
quantitative information for further evaluation.
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FIGURE 5
Illustration of the algorithm used for the image
analysis of the cell deformation in the optical stretcher.
(A) Original phase contrast image of a stretched
spherical RBC. The diameter of the cell is about 6 µm.
(B) The original image mapped onto a rectangle. The
inside of the cell is the lower part of the picture. (C)
Line scan across the cell boundary from top to bottom after squaring
the grayscale values. (D) Line scan after thresholding
and division by the first spatial derivative. (E) The
zigzag line is the boundary of the cell as extracted from the binary
image. The smooth line is the inverse Fourier transform of the three
dominant frequencies in the original data. (F) The white
line shows the image of this smooth line representing the boundary of
the cell as detected by the algorithm converted back into the cell
image. The line matches the cell's boundary to a high degree.
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The algorithm consists of the following steps. First, the
geometrical center of the cell is found and used as the origin of a
polar coordinate system. The grayscale values along the radii outward
from the center are reassigned to Cartesian coordinates (see Fig.
5 B). Essentially, the image is cut along one radius and
mapped onto a rectangle. The bottom of Fig. 5 B is the
interior of the cell and the top is the outside. The cell boundary is
along the wavy line between the black and the white bands. Next, the image is squared to enhance the contrast between cell and background. The effect of this filter can be seen in Fig. 5 C, which
shows a line scan across the cell boundary. If we assume that the
boundary between inside and outside coincides with the first peak, when moving from the outside to the inside of the cell, where the slope of
the line scan is zero, we can drastically enhance the signal-to-noise ratio by dividing the data by their first spatial derivative. Because
this would also enhance peaks in the background noise, we first
threshold the image at a low value to set the background to zero. The
combined effect of this mathematical filter can be seen in Fig.
5 D. The first peak, taken as the cell boundary, is then
easy to detect (zigzag line in Fig. 5 E). The resolution up
to this point is identical to the pixel resolution of the
microscope/CCD system, i.e., 118 ± 2 nm/pixel. To further improve
this resolution, we use the physical constraint that the cell boundary
has to be smooth on this length scale. This is implemented by
Fourier-decomposing the boundary data and by filtering out the high
spatial frequency noise, which increases the resolution to an estimated
±50 nm. The smooth line in Fig. 5 E is the inverse Fourier
transform of the remaining frequencies. The
information about the deformation of cells is then extracted from the
resulting function. Figure 5 F shows the original cell with
the boundary as detected with this algorithm.
In general, this sort of image analysis can yield resolutions down to
±11 nm (see, for example, Käs et al., 1996), which is well below
the optical resolution of the microscope and also below the pixel
resolution. The reason, in short, is that we do not want to resolve two
close-by objects, which is limited to a distance of about half the
wavelength. Instead, the goal is to detect how much an edge,
characterized by a large change in intensity, is moving.
The absolute size determination of an object using this algorithm
depends somewhat on the exact definition of the boundary between object
and surrounding medium in the phase-contrast image. Our choice, as
described above, was driven by the investigation of images of silica
beads with known size. The estimated resolution of ±50 nm is in
agreement with measurements of these beads, which have a radius of
= 2.50 ± 0.04 µm (SD). This resolution is
certainly sufficient to discriminate between stretched and unstretched
cells as reported further below. The advantages of this algorithm are its speed, precision, and its ability to detect the shape of any cell.
While the radii of the cells were measured with the algorithm, the
distances between the fiber tip and the cell, d, were
measured in a simpler way by counting pixels in images. For the 20×
objective, this can be done with a pixel resolution of ±0.6 µm.
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RESULTS AND DISCUSSION |
Theory
Total force for one beam
The simplest way to describe the interaction of light with cells
is by ray optics (RO). This approach is valid when the size of the
object is much larger than the wavelength of the light. The diameter of
cells, 2, is on the order of tens of microns. Cell
biological experiments, such as the optical stretching of cells, are
performed in aqueous solution, and water is sufficiently transparent
only for electromagnetic radiation in the near infrared (the laser used
was operated at a wavelength of = 785 nm). Thus, the
criterion for ray optics, 2
/ 25-130 
1, is fulfilled (van de Hulst, 1957).
The idea is to decompose an incident laser beam into individual rays
with appropriate intensity, momentum, and direction. These rays
propagate in a straight line in uniform, nondispersive media and can be
described by geometrical optics. Each ray carries a certain amount of
momentum p proportional to its energy E and to
the refractive index n of the medium it travels in,
p = nE/c, where c is
the speed of light in vacuum (Ashkin and Dziedzic, 1973; Brevik, 1979).
When a ray hits the interface between two dielectric media with
refractive indices n1 and
n2, some of the ray's energy is
reflected. Let us assume that n2 > n1 and
n2/n1 1, which is the case for biological objects in aqueous media, and
that the incidence is normal to the surface. The fraction of the energy
reflected is given by the Fresnel formulas (Jackson, 1975),
R 103. The momentum of the
reflected ray, pr = n1RE/c, and the
momentum of the transmitted ray, pt = n2(1 R)E/c (Ashkin and Dziedzic, 1973;
Brevik, 1979). The incident momentum,
pi = n1E/c, has to be
conserved at the interface. The difference in momentum between the
incident ray and the reflected and transmitted rays,
p = pi + pr pt, is
picked up by the surface, which experiences a force F
according to Newton's second law,
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(1)
|
where P is the incident light power and Q is
a factor that describes the amount of momentum transferred
(Q = 2 for reflection, Q = 1 for
absorption). For partial transmission of one laser beam hitting a flat
interface at normal incidence as described above, Qfront = 1+ R n(1 R) = 0.086
(n1 = 1.33, n2 = 1.43, n = n2/n1). This force acts in the backward direction, away from the denser medium
(see also Fig. 2). The transmitted ray eventually hits the backside of
the object and again exerts a force on the interface. Here,
Qback = [n + Rn (1 R)](1 R) = 0.094, and the force acts in the forward
direction, again away from the denser medium. For the total force
acting on the object's center of gravity, Qtotal = Qfront + Qback = 0.008. The total force is
obviously an order of magnitude smaller than either one of the surface forces.
If the ray hits the interface under an angle 
0, it changes direction according to Snell's law,
n1sin = n2sin 
, where is the angle of
the transmitted ray. In this case, the vector nature of momentum has to
be taken into account and R becomes a function of the
incident angle . R is taken to be the average of the coefficients for perpendicular and parallel polarization relative to the plane of incidence. This is a negligible deviation from
the true situation (the error in the stress introduced by this
simplification is smaller than 2% for
n2 = 1.45, and smaller than 0.5% for
n2 = 1.38), but it simplifies the
calculation and preserves symmetry of the problem with respect to the
laser axis. The components of the force in terms of Q on the
front side, parallel and perpendicular to the beam axis, are
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(2a)
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and
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(2b)
|
where 
is the angle between the beam axis and the
direction of the momentum transferred. Similarly, on the back the
components of the surface force are
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(3a)
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and
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(3b)
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Subsequent reflected and refracted rays can be neglected because
R < 0.005 for all incident angles. The magnitude of
the force in terms of Q on either the front or the backside
is given by
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(4)
|
which is a function of the incident angle , and the
direction of the force is
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(5)
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The forces on front and back are always normal to the surface for
all incident angles. Thus, the stress 
, i.e., the force per unit area, along the surface where the ray enters and leaves the
cell is
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(6)
|
where I() is the intensity of the light. Figure
6 shows stress profiles calculated for
spherical objects with the refractive index of polystyrene beads, and
with the average refractive index of RBCs hit by one laser beam with
Gaussian intensity distribution. The profiles are rotationally
symmetric with respect to the beam axis. The sphere acts as a lens and
focuses the rays on the back toward the beam axis, which results in a
narrower stress profile. Integrating this asymmetrical stress over the
whole surface yields the total force on the object's center of mass,
which pushes the object in the direction of the beam propagation. At
the same time, the applied stress stretches the object in both
directions along the beam axis. Due to the cylindrical symmetry, the
total force has only a component in the direction of the light
propagation, which is generally called scattering force. If the object
is displaced from the beam axis, this symmetry is broken. It
experiences a force, called gradient force, perpendicular to the axis,
which pulls the object toward the highest laser intensity at the center of the beam if the refractive index of the object is greater than that
of the surrounding medium. Because the gradient force is restoring,
after the object reaches the axis, it will stay there as long as no
other external forces are present and the gradient force is zero.
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FIGURE 6
Surface stress profiles for one laser beam incident
from the left on a polystyrene bead (top row) and a BALB
3T3 fibroblast (bottom row) for different ratios between
the beam radius w and the object's radius
. The radii of the polystyrene bead and the
fibroblast used for this calculation were = 2.55 µm and = 7.70 µm and the refractive indices were
n2 = 1.610 and
n2 = 1.370, respectively. The total
light power P = 100 mW for all profiles. The
concentric rings indicate the stress in Nm2
(note the different scales). The resulting total force after
integration over the surface acting on the center of gravity of the
object is noted in each case.
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The stress profile and the total force depend on the ratio between beam
radius w and sphere radius , and on the
relative index of refraction, n = n2/n1.
If there is only one beam shining on the object, the total force will
accelerate it. Because the beam is slightly divergent (the beam radius
doubles from w = 2.7 µm at the fiber tip to
w = 5.4 µm over a distance of 70 µm), the beam
radius w, and therefore the stress profiles and the total force, are functions of the distance d from the fiber end
(see Fig. 8). Smaller beam size with respect to the object results in
higher light intensity and thus greater stress on the surface (w/ 
1). As the beam radius w
increases with increasing distance d and
w/ approaches one, the light intensity and the
magnitude of the surface stress decrease. However, the total force
increases because the asymmetry between the front and back becomes more pronounced. As the beam size becomes much larger than the object (w/ 1), the surface stresses and the total
force vanish, because less light is actually hitting the object. The
highest total forces (w/ = 1) calculated for
polystyrene beads in water and RBCs in their final buffer for a light
power of P = 100 mW are
Ftotal = 28.6 ± 0.9 pN and
Ftotal = 1.82 ± 0.06 pN,
respectively. The relative error in the force calculations, due to
uncertainties in the measurement of the relevant quantities (indices of
refraction, light power, radius, distance between fiber and object) as
given earlier, is 3.2%. In general, the magnitude of the surface
stress and the total force increase with higher relative indices
of refraction n.
The change in total force as the object is pushed away from the light
source can be measured by setting the accelerating total force
Ftotal equal to the Stokes drag force
acting on the spherical object,
|
(7)
|
where 
is the viscosity of the surrounding medium
and v is the velocity of the object. The viscous drag on a
spherical object can depend strongly on the proximity of boundaries. A
correction factor a can be found in terms of the ratio
between the radius of the sphere and the distance to the
closest boundary b (Svoboda and Block, 1994),
|
(8)
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The Reynolds number is on the order of
104, so inertia can be neglected. The
measurement of the total force on different objects was used to
investigate to what extent cells can be approximated as objects with a
homogeneous index of refraction (see Shooting Experiments).
Stress profiles for two beams
A configuration with two opposing, identical laser beams functions
as a stable optical trap where the dielectric object is held between
the two beams. When the object is trapped, the surface stresses caused
by the two incident beams are additive. Fig.
7 shows the resulting stress profiles for
RBCs, which are rotationally symmetric with respect to the beam axis.
If the object is centered, the surface stresses cancel upon
integration, and the total force is zero. Otherwise, restoring gradient
and scattering forces will pull the object back into the center of the
trap. The trapping force is the minimal force required to pull the
object completely out of the trap, which is equal to the greatest
gradient force encountered as the object is displaced.
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FIGURE 7
Surface stress profiles on an RBC trapped in the
optical stretcher for different ratios between the beam radius
w and the cell radius . The total
power in each beam was P = 100 mW for all profiles.
The radius of the RBC used for this calculation was = 3.30 µm, and the refractive index was
n2 = 1.378. The concentric rings
indicate the stress in Nm2. The peak stresses
0 along the beam axis (0°
and 180° direction) are given below each profile. The trapping of the
cell for w/ < 1 is unstable (see
text). The dashed line for w/ = 1.1 shows the r() = 0cos2() approximation of
the true stress profile.
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Again, the shape of the profile changes depending on n
and the ratio w/. The smaller the beam size
with respect to the object, the greater the stress in the vicinity of
the axis. However, the trap is not stable if the beams are smaller than
the object. This has been discussed and experimentally shown by Roosen
(1977). For w/ 1, the surface stresses
become very small. The ideal trapping situation is when the beams are
only slightly larger than the object (w/ 
1). For example, for w/ = 1.1, the peak stresses 0 along the beam axis for
RBCs trapped with two 100-mW beams,
0 = 1.38 ± 0.05 Nm2. The relative error in the stress
calculations, due to uncertainties in the measurement of the relevant
quantities as given earlier, is 3.0%. In this case, the stress profile
can be well approximated by
r( ) =
0cos2()
(see Fig. 7 for w/ = 1.1). This functional
form of the stress profile makes an analytical solution of the
deformation of certain elastic objects tangible.
Deformation of thin shells
Erythrocytes were used initially because they are soft, easy to
handle and to obtain, and their deformations are easily observed. They
are also much more accessible to theoretical modeling than eukaryotic
cells with their highly complex and dynamic internal structures. Thus,
RBCs can be considered well-defined elastic objects that can be used to
verify the calculated stress profiles. The only elastic component of
RBCs is a thin composite shell made of the plasma membrane, the
two-dimensional cytoskeleton, and the glycocalix. The ratio between
shell radius and shell thickness h,
/h 100. In this case, membrane theory
can be used to describe deformations due to surface stresses
(Mazurkiewicz and Nagorski, 1991; Ugural, 1999).
Membrane theory is the simplification of a more general theory of the
deformation of spherical shells in which the bending energy
Ub of the shell is neglected and only
the membrane (or stretching) energy Um
is considered. It can be shown that the ratio of those two energies for
the case of axisymmetric stress, as applied with the optical stretcher,
is
Ub/Um = 4h2/32 104 for RBCs. The stress
r applied to a spherical object in
the optical stretcher has the form,
r = 0cos2(),
as shown above. Spherical coordinates are an obvious choice, where the
radial direction is denoted by r, the polar angle by 
, and the azimuthal angle is . The
coordinate system is oriented such that the incident angle of the rays,
, in the previous section is identical to the polar angle
(the laser beams are traveling along the
z-axis). The total energy U of a thin shell
consists of the membrane energy and the work done by the stress applied and is given by,
|
(9)
|
where 
and
are the strains in the polar
and meridional direction, respectively,
ur is the radial deformation of the
membrane, E is the Young's modulus, and 
is
the Poisson ratio. The connections between the strains and the
deformations are
|
(10a)
|
and
|
(10b)
|
where u is the deformation in
meridional direction. The radial and meridional displacements, which
describe the experimentally observed deformation of the dielectric
object in the optical stretcher, can be found by using Euler's
equations,
|
(11a)
|
and
|
(11b)
|
where F is the integrand of the energy functional,
|
(12a)
|
and
|
(12b)
|
Using Eqs. 10, 11, and 12 and the explicit form of
r(), we find the
following expressions for the radial and the meridional deformations of
the membrane,
![u<SUB><UP>r</UP></SUB>(&thgr;)=<FR><NU>&rgr;<SUP>2</SUP>&sfgr;<SUB>0</SUB></NU><DE>4Eh</DE></FR> [(5+v)<UP>cos</UP><SUP>2</SUP>(&thgr;)−1−v]](/content/vol81/issue2/fulltext/767/img032.gif)
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(13a)
|
and
|
(13b)
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As expected from the symmetry of the problem, the deformations
ur and
u are independent of the azimuthal
angle , and there is also no displacement
u in this direction. Figure 11 shows
the shapes of thin shells calculated with Eq. 13a for = 0.5, which is normal for biological membranes, and Eh = (3.9 ± 1.4) × 105
Nm1, which was found to be the average for RBCs
from the experiments (see below) for increasing stresses
0. Because these equations are
linear, they only hold for small strains (<10%). In the microscope images, which are cross-sections of the objects because the focal depth
of the objective is much smaller than the diameter of the RBCs, only
the radial deformations can be observed. The direct comparison between
the theoretically expected deformations
ur() and the
experimentally observed radial deformations help to establish the RO
model as valid explanation for the optical stretching of soft
dielectric objects.
Experiments
Shooting experiments
To test the assumptions underlying the RO calculations as
described above, we measured the total force acting on different objects. It was not clear if it was permissible to model living cells
with their organelles and other small-scale structures as homogeneous
spheres with an isotropic index of refraction. Although this assumption
is obvious for RBCs homogeneously filled with hemoglobin, it might be
questionable in the case of eukaryotic cells containing organelles and
other internal structures (see Fig. 3). In this series of experiments,
individual silica beads, polystyrene beads, or fibroblasts were trapped
in the optical stretcher. The setup was identical to the one used for
the trapping and stretching of RBCs (see Experimental Setup and Fig.
4). After stably trapping the objects, we blocked one of the laser
beams. The total force from the other beam accelerated the object away from the light source.
The total force was determined using Eqs. 7 and 8 and the velocities,
radii, and distances measured during the experiment. In our setup, the
distance b between the moving objects and the coverslip as
closest boundary was b = 62.5 ± 2.5 µm (half
the diameter of the optical fiber). In the case of the silica and polystyrene beads, this distance is about 25 times the radius of the
beads and the correction factor a = 1.023. For the
cells, the distance is ~7-9 times the radius and a = 1.072-1.090. The viscosity used in the calculation was
that for water at 25°C, = 0.001 Pa s. Figure
8 shows the total force measured as a
function of the distance d between the fiber tip and the
object, as well as the total force as expected from our RO
calculations. The error bars shown are statistical errors in the
experimental data. The relative errors introduced by the uncertainties
in the measurements of radii, velocities, and distances are negligibly
small (1.1-3.1%). It is not surprising that the experimental data
points for silica beads and polystyrene beads match the theory because
these are truly spherical and have an isotropic index of refraction.
The fact that, also, the experimental results for the cells matched the
theory was proof that even eukaryotic cells can be treated using this
simple RO model. The magnitude and the dependence of the total force on
the distance d are in good agreement with the results by
(Roosen, 1977).
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FIGURE 8
The total forces from one laser beam on silica beads,
polystyrene beads, and BALB 3T3 fibroblasts as functions of the
distance d between the fiber tip and the object. The
data points are from measuring the total force on the objects in the
optical stretcher after obstructing one of the laser beams. The solid
lines represent the total forces calculated using ray optics. The radii
of the silica beads, the polystyrene beads, and the fibroblasts were
= 2.50 ± 0.04 µm, = 2.55 ± 0.04 µm, and = 7.70 ± 0.05 µm, and the refractive indices were
n2 = 1.430 ± 0.003, n2 = 1.610 ± 0.005, and
n2 = 1.370 ± 0.005, respectively.
The total light power P is indicated in each case. The
error bars represent standard deviations; the error in the distance
measurement was ±0.6 µm.
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Stretching of erythrocytes
Single, osmotically swollen RBCs were trapped in the optical
stretcher at low light powers (P 5-10 mW). The
light power was then increased to a higher value between
P = 10 and P = 800 mW for approximately
5 s, an image of the stretched cell was recorded, and the light
power was decreased again to the original value. During the short time
intervals when stress was applied, we did not observe any creep, i.e.,
any increase in deformation during the duration of the stretching.
Also, the cells did not show any hysteresis or any kind of plastic
deformation up to P 500 mW. Figure
9 shows a sequence of RBC images recorded
at increasing light powers. It is obvious that the deformation
increased with the light power used. The radius along the beam axis
increased from 3.13 ± 0.05 µm in the first image to 3.57 ± 0.05 µm in the last image, a relative increase of 14.1 ± 0.3%, whereas the radius in the perpendicular direction decreased from
3.13 ± 0.05 µm to 2.77 ± 0.05 µm (relative change
11.5 ± 0.2%).
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FIGURE 9
Typical sequence of the stretching of one osmotically
swollen RBC for increasing light powers. The top row shows the RBC
trapped at 5 mW in each beam. The power was then increased to the
higher power given below, which lead to the stretching shown
underneath, and then reduced again to 5 mW. The stretching clearly
increases with increasing light power. The radius of the unstretched
cell was = 3.13 ± 0.05 µm, and the distance
between the cell and either fiber tip was d 60 µm. The images were obtained with phase contrast microscopy. Any
laser light was blocked by an appropriate filter.
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From the radius of the cell, the distance between cell and fiber tips,
the refractive indices of cell and medium, and the power measured, we
calculated the stress profiles for each cell using the RO model. As
mentioned earlier, the relative error in the stress calculation is
3.0%. The peak stress in the last image of Fig. 9 was calculated as
0 = 1.47 ± 0.03 Nm2. Figure 10
shows the relative increase in radius along the beam axis
( = 0) and the relative decrease perpendicular
( = /2) versus the peak stress
0 for 55 RBCs. The error bars shown
are statistical errors. The relative errors in the calculation of the
relative changes and the peak stresses due to uncertainties in the
relevant quantities measured are comparatively small. The solid line in
Fig. 10 shows a fit of
ur(0)/ (see Eq. 13a) to
the experimental data. For the fit, the errors in the peak stresses were neglected and a singular value decomposition algorithm was used.
The data points were weighed with the inverse of the standard deviations. The resulting slope was 0.080 ± 0.011 m2N1, which yielded
Eh = (3.9 ± 1.4) × 105 Nm1. The intercept
was zero. The correlation coefficient for the fit was,
r = 0.92, excluding the last data point. This shows
that up to 0 2 Nm2 (P 350 mW) and relative
deformations of about 10%, the response of the RBCs was linear. In
this regime, linear membrane theory can be used to describe the
deformation of RBCs.
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FIGURE 10
Relative deformation
ur/ of RBCs along
(positive values) and perpendicular (negative
values) to the laser axis in the optical stretcher as a
function of the peak stress 0. The error
bars for the relative deformations and the peak stresses are standard
deviations. The solid line shows a fit of Eq. 13a as derived from
membrane theory to the data points using a singular value decomposition
algorithm. The linear correlation coefficient for this fit,
r = 0.92, (excluding the last data point) indicates
a linear response of the RBCs to the applied stress. Beyond a peak
stress 0 2 Nm2 the deformation starts deviating from
linear behavior.
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In the literature, usually the cortical shear modulus Gh is
given, rather than the Young's modulus Eh. The quantities
are related by Gh = Eh/2(1 + ) = Eh/3. In our case, the shear modulus, Gh = (1.3 ± 0.5) × 105 Nm1. This value is
in good agreement with values reported previously from micropipette
aspiration measurements, which yielded shear moduli in the range
6-9 × 106 Nm1
(Hochmuth, 1993). Micropipette aspiration is the most established technique for measuring cellular elasticities, and the value for the
shear modulus of the RBC membrane has been confirmed many times and is
well accepted.
More recently, optical tweezers were used to measure the shear modulus
with very differing results. In these experiments, beads were attached
to the membrane on opposite sides of the RBC, trapped with optical
tweezers, and then displaced. The values found ranged from (2.5 ± 0.4) ×106 Nm1
(Hénon et al., 1999) to 2 × 104
Nm1 (Sleep et al., 1999). Because this
technique applies point forces to the membrane, the stress is highly
localized and leads to nonlinear deformations. The discrepancy between
these values and the established values for the shear modulus can
probably be attributed to this different load condition.
Furthermore, the theoretically expected and the observed shapes of RBCs
in the optical stretcher coincide well (see Fig.
11). The white lines are the shapes of
thin shells with RBC material properties as predicted by linear
membrane theory subjected to the surface profile calculated by the RO
model. These lines were overlaid on the images of the stretched RBCs in
Fig. 9. The excellent agreement between the predicted and the observed
shaped shows that using RO theory is sufficient to calculate the
surface stress on cells in the optical stretcher. An ab inito treatment
of the interaction of a spherical dielectric object in an inhomogeneous electromagnetic wave using Maxwell's equations and surface stress tensor would be much more difficult and is also not necessary in this
case. The RO model is powerful enough to accurately predict the
qualitative and the quantitative aspect of the stretching. Ray optics
has the additional benefit of being much more accessible.
¶
and
TOP
ABSTRACT
INTRODUCTION
![−(1−R(&bgr;))<UP>cos</UP>(2&agr;−2&bgr;)]](/content/vol81/issue2/fulltext/767/img014.gif)
![−(1−R(&bgr;))<UP>sin</UP>(2&agr;−2&bgr;)]](/content/vol81/issue2/fulltext/767/img018.gif)
