Biomedical Engineering Reference
In-Depth Information
Fig. 14.11.
A schematic diagram of optical forced oscillation via (
a
) oscillatory
optical tweezers and (
b
) a set of twin optical tweezers
(Fig. 14.11b). In both cases, the steady-state oscillation amplitude and the
relative phase (with respect to that of the driving source) of the oscillating
particle can be conveniently measured with the aid of a quadrant photo-diode
in conjunction with a lock-in amplifier. By changing the driving frequency,
typically in the range of approximately a few hertz to a few hundred hertz,
the oscillation amplitude and the relative phase of the oscillating particle can
be plotted as a function of frequency. In general, the experimental data fit
fairly nicely with the theoretical results deduced from a simple theoretical
model of forced-oscillation with damping [48, 50]. By coating a microparticle
with a protein of interest and allowing the oscillating particle to interact with
protein receptors on a cellular membrane, the method described above can be
used to measure the protein-protein interaction, which can be modeled as an-
other linear spring in parallel to the optical spring (Fig. 14.12). The equation
of motion of a particle, suspended in a viscous fluid, trapped, and forced to
oscillate in oscillatory optical tweezers, can be written as
−β x − k
x
(
t
)
− A
e
i
ωt
,
mx
=
(14.8)
where
m
is the mass of the particle,
β x
=6
πηr x
is the viscous drag prescribed
by Stokes' law,
η
is the coe
cient of viscosity of the surrounding fluid,
r
is
the radius of the particle,
k
is the optical spring constant in the linear spring
model, and
A
and
ω
are the amplitude and the frequency of the focal spot of
the oscillatory optical tweezers, respectively.
In the case of a set of twin optical tweezers with the particle interacting
with a cell as is illustrated schematically in Fig. 14.12, the equation of motion
of the particle can be written as [48]
k
(e
i
ωt
+1)
2
mx
=
−β x −
(
x − x
1
)
−
(
k
+
k
int
)(
x − x
2
)
,
(14.9)
where
x
1
and
x
2
represent the position of the force center of optical tweezers 1
and 2, respectively;
ω
is the fundamental harmonics of the chopping frequency;
