Biology Reference
In-Depth Information
Variation of the magnetic field (and thus magnetic force) at the sample plane is
achieved by vertically translating the magnets with a DC-servo motor (such as model
M-126.PD1; Physik Instrumente).
Separation distances between the sample plane and magnet range from near con-
tact with the coverslip to tens of millimeters away. The conversion from separation
distance to force along the optical axis depends on the magnet type, number, and
geometry. Although it is possible to use finite element (FE) modeling to predict with
reasonable accuracy the magnetic field generated by a specific magnet array, it is
difficult to determine the magnetic moment of commercial superparamagnetic
beads, which depends on both the bead properties and the magnitude of the applied
magnetic field. Thus, it is typically more reliable to determine experimentally the
relationship between separation distance and applied force.
6.1.3.2
Calibrations
Two calibration methods are common (
Kim & Saleh, 2008; Manosas et al., 2010;
Ribeck & Saleh, 2008
). In the first, forces are calibrated by measuring the Brownian
motion of a magnetic bead that is tethered to the coverslip by a single DNAmolecule,
and thus acts as a simple inverted pendulum. The lateral spring constant is given by
the ratio of the vertical force to the DNA length. This spring constant is determined
by modeling the measured bead trajectory with an overdamped Langevin equation
of motion for a particle in a harmonic potential, and fitting the measured power
spectrum in position to that predicted from the Langevin model after accounting
for issues of finite data sampling rate and instrumental low-pass filtering.
The best-fit spring constant, along with the measured length, gives an estimate of
the force; this calibration is then repeated at each desired magnet position. This
method is particularly useful at small forces, but the DNA tethers tend to break at
forces above
60 pN (
Fig. 6.5
).
For larger forces, or in cases where purified DNA molecules of known length are
not readily available, it is preferable to measure the velocity
v
at which single mag-
netic beads of radius
R
move through a solution of known viscosity
, then to relate
this velocity to the force
F
using Stokes law
F
Rv
. The range of velocities that
can be reliably determined depends on the camera frame rate and the maximum dis-
tance over which particle position can be reliably tracked. At a minimum,
¼
6
p
4-6 par-
ticle positions must be determined to accurately measure velocity. We have reliably
measured bead velocities over the range of
m/s using a pure glycerol
solution and a fast CMOS camera with 280 frames per second (fps) for full-frame
collection (
Lin & Valentine, 2012a
). Although tables of glycerol viscosity are avail-
able, we recommend independent verification, since viscosity depends sensitively on
temperature, and glycerol solutions tend to absorb water in humid environments,
which will decrease glycerol concentration and viscosity. We determined viscosity
of our glycerol sample to be
1.15 Pa s, using a strain-controlled rheometer (ARES-
LS, TA Instruments) with a cone-plate tool geometry (50 mm diameter, 0.04 rad
cone angle, and 0.045 mm gap) at
0.005-300
m
19
C and a strain rate of 1 s
-1
.
