An Introduction to Magnetic Tweezers

Abstract

Magnetic tweezers are a single-molecule force and torque spectroscopy technique that enable the mechanical interrogation in vitro of biomolecules, such as nucleic acids and proteins. They use a magnetic field originating from either permanent magnets or electromagnets to attract a magnetic particle, thus stretching the tethering biomolecule. They nicely complement other force spectroscopy techniques such as optical tweezers and atomic force microscopy (AFM) as they operate as a very stable force clamp, enabling long-duration experiments over a very broad range of forces spanning from 10 fN to 1 nN, with 1–10 milliseconds time and sub-nanometer spatial resolution. Their simplicity, robustness, and versatility have made magnetic tweezers a key technique within the field of single-molecule biophysics, being broadly applied to study the mechanical properties of, e.g., nucleic acids, genome processing molecular motors, protein folding, and nucleoprotein filaments. Furthermore, magnetic tweezers allow for high-throughput single-molecule measurements by tracking hundreds of biomolecules simultaneously both in real-time and at high spatiotemporal resolution. Magnetic tweezers naturally combine with surface-based fluorescence spectroscopy techniques, such as total internal reflection fluorescence microscopy, enabling correlative fluorescence and force/torque spectroscopy on biomolecules. This chapter presents an introduction to magnetic tweezers including a description of the hardware, the theory behind force calibration, its spatiotemporal resolution, combining it with other techniques, and a (non-exhaustive) overview of biological applications.

Full text

Chapter 18
An Introduction to Magnetic Tweezers
David Dulin
Abstract
Magnetic tweezers are a single-molecule force and torque spectroscopy technique that enable the mechani-
cal interrogation in vitro of biomolecules, such as nucleic acids and proteins. They use a magnetic field
originating from either permanent magnets or electromagnets to attract a magnetic particle, thus stretching
the tethering biomolecule. They nicely complement other force spectroscopy techniques such as optical
tweezers and atomic force microscopy (AFM) as they operate as a very stable force clamp, enabling
long-duration experiments over a very broad range of forces spanning from 10 fN to 1 nN, with
1–10 milliseconds time and sub-nanometer spatial resolution. Their simplicity, robustness, and versatility
have made magnetic tweezers a key technique within the field of single-molecule biophysics, being broadly
applied to study the mechanical properties of, e.g., nucleic acids, genome processing molecular motors,
protein folding, and nucleoprotein filaments. Furthermore, magnetic tweezers allow for high-throughput
single-molecule measurements by tracking hundreds of biomolecules simultaneously both in real-time and
at high spatiotemporal resolution. Magnetic tweezers naturally combine with surface-based fluorescence
spectroscopy techniques, such as total internal reflection fluorescence microscopy, enabling correlative
fluorescence and force/torque spectroscopy on biomolecules. This chapter presents an introduction to
magnetic tweezers including a description of the hardware, the theory behind force calibration, its
spatiotemporal resolution, combining it with other techniques, and a (non-exhaustive) overview of
biological applications.
Key words Single-molecule biophysics, Magnetic tweezers, Force spectroscopy, Protein-nucleic acids
interactions, Torque spectroscopy
1 Brief History and Application of Magnetic Tweezers
Magnetic tweezers use the magnetic field generated by permanent
or electromagnets to apply force and/or rotate magnetic particles
attached to a biological material, hence inducing a mechanical
stress. The first biophysics assay using a magnetic actuator in a
biological context was reported by Crick and Hughes in 1950
[1], where magnetic particles placed in the cytoplasm of a cell
were displaced to interrogate its viscoelastic properties. Magnetic
tweezers have two main areas of application in biophysics: cellular
mechanics and single-molecule biophysics. The force applied in
Iddo Heller et al. (eds.), Single Molecule Analysis: Methods and Protocols, Methods in Molecular Biology, vol. 2694,
https://doi.org/10.1007/978-1-0716-3377-9_18,
© The Author(s) 2024
375

cellular mechanics is usually relatively large (≫ 1 nN) [2, 3] com-
pared to the single-molecule world (≪100 pN) [4], and therefore
their respective instrument designs differ significantly. Here, we
solely focus on magnetic tweezers assays for single-molecule
biophysics.
376 David Dulin
In the 1990s, the Bustamante lab and the Croquette and
Bensimon lab pioneered the modern version of single-mole-
cule magnetic tweezers capable of applying a constant force (even
below 1 pN) and torque, enabling the interrogation of DNA
mechanical properties at the single-molecule level [5,6]. Nowadays,
magnetic tweezers are found in many labs around the world, and
single-molecule studies have been performed on various protein-
nucleic acids systems [7], ranging from helicases [8] to DNA poly-
merases [7–9], topoisomerases and gyrases [10–13], cellular and
viral RNA polymerases [14], nucleoprotein filaments [15–18], and
the mechanical stability of protein folding and protein-ligand inter-
actions [19–23]. The simplicity and the robustness of the technique
make it a powerful single-molecule force spectroscopy assay that
has become more and more popular in the academic community.
2 Description of a Magnetic Tweezers Apparatus
Magnetic tweezers for single-molecule studies are composed of a
collimated light source located above a magnetic field source (e.g.,
permanent magnets) that is mounted on top of a flow cell in which
super-paramagnetic beads (simply coined magnetic beads from
now on) are tethered to the flow cell coverslip surface by a biomol-
ecule (Fig. 1a). The magnetic beads are imaged using an inverted
microscope onto a camera, which enables the tracking of their
three-dimension position as a function of time. The latest comple-
mentary metal-oxide-semiconductor (CMOS) cameras enable both
high-throughput (Fig. 1b)[24–27] and high-speed measurements
[28–30]. The vertical and angular position of the magnets is
adjusted using linear motors to vary the force (Fig. 1c) and the
torque (Fig. 1d), respectively, applied to the biomolecule. The
inverted microscope body may be either custom build or bought
commercially. Given its simple design, the custom body presents
only a mild difficulty to produce and is mechanically more stable.
2.1 Magnet
Configuration
Different configurations have been used in magnetic tweezers
experiments to modulate how force and torque are applied
[15]. The most standard configuration uses a pair of cubic perma-
nent magnets, being either vertically (Fig. 1a, c, and d) or horizon-
tally aligned [31]. While the gap between the magnets gives access
to the light source, it also modulates the applied force: a smaller
gap results in a larger maximum force but reduces the surface
area that experiences a homogenous force field [32, 33]. The

two-permanent magnet cubes configuration strongly clamps the
magnetic bead in rotation, which precludes specific applications,
e.g., directly measuring the torsional stiffness of a soft biomolecule
such as DNA. Alternative magnet geometries have therefore been
developed using either a single cylindrical magnet [34] or addition-
ally having a small side magnet attached to it to angularly trap the
magnetic bead mildly [35, 36]. These methods have been reviewed
in detail in Ref. [15].
An Introduction to Magnetic Tweezers 377
Fig. 1 Magnetic tweezers for single-molecule applications. (a) Schematic of a magnetic tweezers instrument
(Adapted from Ref. [61]). (b) Field of view of a high-throughput magnetic tweezers assay, where ~500
magnetic beads of 2.8 μm diameter can be followed simultaneously (50× magnification, 120 nm pixel size)
(Adapted from Ref. [49]). (c, d) Force and torque spectroscopy, respectively, of a single nucleic acid molecule
using magnetic tweezers. The nucleic acid molecule is attached to the magnetic bead via a biotin-streptavidin
bond, and to the surface via digoxigenin-antidigoxigenin attachment. Biotin and digoxigenin molecules are
inserted nonspecifically during the synthesis of the nucleic acid handles that are subsequently ligated to the
main nucleic acid strand (Adapted from Ref. [51])

378 David Dulin
2.2 Illumination Different light sources have been used to illuminate the sample. A
source is chosen to be spatially and temporally coherent, which
generates many diffraction rings with a good contrast to enable
an excellent tracking resolution. Light-emitting diodes (LEDs) are
a simple solution that provide a good temporal coherence (~10 nm
spectral dispersion, full width at half maximum), are easily colli-
mated using a high numerical aperture aspherical lens, and provide
enough light intensity to image the beads in standard image acqui-
sition frequency (50–100 Hz) (Fig. 1a). To achieve high-speed
image acquisition (≥ kHz), a high photon flux through the sample
is required. Unfortunately, LEDs can hardly satisfy this require-
ment. With a noncoherent light source, a high flux may be achieved
using a fiber-coupled arc lamp in combination with a spectral filter
[30, 37]. Coherent sources such as laser diodes and super-
luminescent diodes also enable an efficient collection and collima-
tion of the output light onto the sample [38]. This enables short
camera shutter times and therefore high image acquisition rates.
However, these coherent sources have the shortcoming of creating
spurious speckle patterns in the field of view. Dark field illumina-
tion, i.e., by blocking the zero-order light pathway, has recently
demonstrated the best to date resolution by reducing the back-
ground noise significantly [39].
2.3 Bead Position
Tracking Algorithm
To follow biomolecular reactions with magnetic tweezers, one must
precisely track the magnetic bead’s position in three dimensions. To
this end, different algorithms have been developed, all using the
diffraction pattern originating from the out-of-focus micron-sized
beads (Fig. 2a–c). A region of interest around the bead is defined a
priori to indicate which area of the camera image contains single,
insolated beads (Fig. 2a, b). A lookup table is acquired before the
start of the experiment by capturing a diffraction pattern of the bead
at different objective positions along the z-axis (Fig. 2a)
[40, 41]. The objective is displaced using a high-resolution piezo
stage in steps of ~50–100 nm (Fig. 1a). To determine the bead’s
position in the (x,y)-plane, a rough estimate is obtained from a center
of mass, followed by a cross-correlation algorithm (Fig. 2d). More
sophisticated versions of this tracking algorithm have been devel-
oped, such as the quadrant interpolation method [25, 42]. To deter-
mine the axial position (z-axis), the diffraction pattern of the
magnetic bead at any given frame is compared to the lookup table
using a squared error metric. Sub-plane resolution is then obtained
by a polynomial fit of the resulting error curve (Fig. 2a). Because of
the simplicity of these tracking algorithms, hundreds of beads may be
followed in parallel and in real-time using modern GPUs to perform
the calculation [25, 30].

An Introduction to Magnetic Tweezers 379
Fig. 2 Bead localization in a magnetic tweezers experiment. (a) Left, diffraction
pattern of a 3 μm diameter reference bead attached to the surface of a coverslip
at different distances from the bead to the microscope objective’s focal plane
along the z-axis. Right, a lookup table built up from radial intensity profiles
across the center of the images of the dif fraction pattern taken at different focal
plane positions spaced by 50 nm intervals. (b, c) Diffraction pattern and intensity
profile along the x-axis (black line). (d) Autocorrelation function (ACF) between
two profiles as in (b) separated by 10 pixels (gray). The maximum of ACF is
indicated by the arrow

380 David Dulin
2.4 Temperature
Control
Enzymatic reactions are sensitive to temperature fluctuations and
follow the Arrhenius law:
kTðÞ=Ae -Ea=kBT ,ð1Þ
where k is the forward reaction rate constant, A is a pre-exponential
factor, E
a
is the activation energy, k
B
the Boltzmann constant, and
T the temperature [43]. It is therefore of great importance to
precisely control the temperature in the flow chamber. Several
articles have been reported on establishing a temperature control
on the flow chambers [44–46]. Simulation and data have clearly
demonstrated that the main heat sink is the oil immersion objective
[45], which directly contacts with the glass coverslip area where the
reaction occurs. In conclusion, controlling the temperature at the
objective enables a precise control of the temperature of the reac-
tion (±0.1 °C), which can be achieved using a simple device com-
mercially available from Thorlabs [46].
2.5 Surface
Functionalization and
Nucleic Acid Construct
Fabrication
Magnetic tweezers are a surface-based technique (Fig. 1a); there-
fore, the flow chamber’s glass surface should be treated with care to
prevent nonspecific attachment to the surface of either the mag-
netic beads or the biomolecules of interest. Different types of
surface functionalization have been developed, such as polyethyl-
ene glycol (PEG) [47, 48], nitrocellulose [49], and lipid bilayer
[50]. The last two are being described in detail in Chapter 21. The
type of attachment is of great importance and therefore defines the
methodology to generate the tether. The standard method of
tethering the magnetic beads relies on fabricating nucleic acids
containing both a biotin handle on one end, to attach the strepta-
vidin coated bead, and a digoxigenin (dig) handle on the other end
to attach the nucleic acid to anti-digoxigenin (anti-dig) antibodies
adsorbed to the flow chamber’s glass surface [51]. Such labels are
introduced when generating the nucleic acid by adding dig- or
biotin-labeled UTP to the nucleotide sets. While biotin-
streptavidin forms a very stable bond, the dig-anti-dig bond is
much weaker and not suitable for high-force or long experiments,
even when using glutaraldehyde to cross-link proteins to the nitro-
cellulose surface [52]. For such experiments, covalent attachment is
preferred to replace the dig-anti-dig bond using either a PEG
functionalized surface with covalent chemistry to attach the bio-
molecule to the surface [47, 48] or a direct attachment [ 20],
providing a tether with a much longer tether surface attachment
lifetime .
DNA and RNA construct fabrication rely either on specific
ligation of double-stranded ends (Fig. 3a, b), annealing single-
stranded nucleic acids, or a combination of both. Very detailed
protocols can be found in several method articles [51–56], and
this topic will therefore not be further discussed here.

An Introduction to Magnetic Tweezers 381
Fig. 3 DNA construct fabrication for single-molecule force spectroscopy experiments. (a) Steps in synthesizing
double-stranded DNA constructs. A plasmid is digested to generate a stem. Handles labeled with either biotins
(BIO) or digoxigenins (DIG) are generated by PCR using λ phage DNA as a template and by adding either
bio-dUTP or dig-dUTP in the reaction solution. The handles are purified, digested, and ligated to the stem. (b)
Similar approach as in (a) to fabricate a DNA hairpin. The different segments are produced by PCR digestion
and ligated together to shape as a hairpin
3 Physical Principles
3.1 Force and Torque
Origin
Magnetic tweezers can apply forces between a femto-Newton
(fN) and a nano-Newton (nN) [4], which depends on the magnetic
bead size (i.e., the total amount of magnetic content) and the
magnet configuration. In the configuration described in Fig. 1a,
reducing the gap between the two magnets increases the force.
Because of the very large force range accessible, magnetic tweezers
have been applied to investigate very different biomolecular sys-
tems. The force experienced by a magnetic particle in a magnetic
field is described by:
F
→
mag = 1
2 ∇
→
m
→
sat B
→
,ð2Þ
→
where F mag is the magnetic force, m
→
sat is the saturated magne-
tization of the particle, and B
→
is the magnetic field [32]. Interest-
ingly, the magnetic force is directly proportional to the gradient of
the magnetic field, not its magnitude.
One of the key aspects of magnetic tweezers is their ability to
apply torque to the tether [6, 57], and torque spectroscopy was
performed on many different complexes, e.g., double-stranded

nucleic acids and nucleoprotein filaments [15]. The magnetic beads
are made of super-paramagnetic nanoparticles embedded in a latex
matrix, and therefore their magnetization should align with the
magnetic field. However, an asymmetry in the nanoparticle spatial
organization induces an anisotropy in m
→
sat, with a minor compo-
nent m
→
0 not aligned with B
→
[58]. This induces a torque Γ
→
on the
bead, which is derived from:
382 David Dulin
Γ
→
= m
→
0 × B
→
:ð3Þ
The torque response of the biomolecule is negligible in respect
to Γ
→
. Hence, rotating the magnets induces a rotation of the mag-
netic bead, thereby transferring torque to the coilable tether. An
example of coilable tether is a fully double-stranded nucleic acid
molecule with multiple attachment points at both ends (Fig. 1a).
To measure the torque response of the biomolecule, and therefore
its torsional stiffness and related mechanical properties, different
magnet configurations have been developed to reduce the magni-
tude of Γ
→
, such as the magnetic torque tweezers [35].
3.2 Force Calibration The force F
mag
may be calibrated from Eq. (1), by using m
sat
(from
the factory specifications of the magnetic beads) and spatially char-
acterizing the magnetic field generated by the magnets with a Hall
probe [32]. However, this method is not the preferred one, as it
relies on parameters measured externally for a given batch of beads.
Therefore, in situ force calibration methods that rely on parameters
directly measured in the magnetic tweezers assay are preferred
[59]. To this end, the theory relating the force as a function of
the tethered magnetic bead lateral fluctuations is derived below.
The force may therefore be extracted from measuring such
fluctuations.
The position of a tethered magnetic bead experiencing a force
F
mag
is best described as an inverted pendulum (Fig. 4a, b)[6, 60]
with two representative cases of pendulum lengths coined short
(Fig. 4a) and long pendulum (Fig. 4b), respectively. In the former
case, the fluctuation in position along the x-axis is pinned by the
magnetic field B
→
(Fig. 4a), and the length of the pendulum is
therefore the length of the tether L
ext
. In the latter case, the
fluctuation in position of the bead along the y-axis is not con-
strained by the magnetic field (Fig. 4a), and the length of the
pendulum is therefore L
ext
+ R, where R is the magnetic bead
radius.
Considering the short pendulum case, the small displacement
δx from the equilibrium position caused by the collisions with the
water molecules (Fig. 4c), i.e., the Brownian motion, induces a
restoring force that can be described as:
Frestoring =kxδx,ð4Þ

h i
An Introduction to Magnetic Tweezers 383
Fig. 4 Force calibration in a magnetic tweezers instrument. (a, b) Schematic of the tethered magnetic bead
position fluctuations along either (a) the x-axis (short pendulum) or (b) the y-axis (long pendulum). (c)
Schematic of the forces exerted on the magnetic bead in the short pendulum configuration. (d) Position of
a magnetic bead along the x-axis against the y-position (left) and time (right). Data taken at three different
forces. (e) Force calibration for M270 (blue, 2.8 μm diameter) and MyOne (red, 1 μm diameter) magnetic
beads as a function of the distance of the magnets to the flow chamber (Adapted from Ref. [61])
where k
x
is the trap stiffness along the x-axis. Therefore,
Frestoring =Fmag sin θðÞ=Fmag δx=Lext,ð5Þ
and
kx =Fmag =Lext,ð6Þ
where θ is the angle spanned by the tether when the bead is at its
current and equilibrium positions, and L
ext
is the length of tether.
At small θ, the potential energy landscape U
x
is quadratic [60], i.e.,
Ux = 1=2ðÞkxδx2 ,ð7Þ
and we can therefore apply the equipartition theorem
( U
x
= (1/2) ∙ k
B
T) on Eq. (7), and we obtain:

ÞÞ
ÞÞ
384 David Dulin
hδx2i=kBT=kx =kBTLext=Fmag ,ð8Þ
and, by extension, for the long pendulum case:
δy2 =kBT=ky =kBTLext þ RðÞ=Fmag:ð9Þ
Equation (8) and (9) directly link the applied force with the
fluctuations in the lateral position of the bead and the tether length
(Fig. 4d). Both are parameters that one can easily retrieve from
experiments to enable a direct force calibration as a function of the
distance of the magnets from the magnetic bead (Fig. 4e)[61, 62].
To provide an accurate force calibration, the lateral fluctuation
of the bead must be measured accurately to not overestimate the
force (Eqs. 8 and 9). To this end, one should make sure the image
acquisition does not overly integrate (meaning average) the mag-
netic bead position fluctuation [61]. From the equation of motion
of the magnetic bead experiencing F
mag
, we are able to extract the
characteristic time scale of the bead:
tc,x =γ=kx =6πηRLext=Fmag and tc,y =γ=ky
=6πηRL
ext þ RðÞ=Fmag,ð10Þ
where γ is the drag coefficient, η the viscosity of the solution
(typically water, i.e., ~10
-3
Pa.s), and R the radius of the magnetic
bead and defines the time during which the bead has explored the
trap. For L
ext
≤ R, the drag coefficient must be corrected to include
the effect of the surface, as described by the Faxe
´n law [63]:
γFaxen =6πηR= 1-9=8 R= RþLext
ðÞðÞþ1=2 R= RþLext
ðÞðÞ
3-57=100 R= RþLext
ðð4
þ1=5 R= RþLext
ðÞðÞ
5þ7=200 R= RþLext
ðÞðÞ
11 -1=25 R= RþLext
ðð12 ,
ð11Þ
hδx
2
i averages away toward a measured value hδx
2
i
meas
as a
function of the camera shutter time τ
sh
and t
c, x
as
δx2
meas = 2kBT=πkx
ðÞarctan 4πtc,x=τsh :ð12Þ
To minimize the error in the force due to camera image blur-
ring, we must minimize the difference between hδx
2
i and hδx
2
i
meas
.
For example, to measure F
mag
with a 10% error due to camera
image blurring, τ
sh
must be at least four times smaller than t
c, x
[61]. How feasible is this in practice? Most large chip CMOS
cameras acquire images with a frequency f
ac
~10 -100 Hz, while
the characteristic time for a L
ext
= 1 μm, R = 1.4 μm and
F
mag
= 10 pN (typical experimental conditions) is t
c, x
~0.03 s,
i.e., similar to τ
sh
for zero-dead time image acquisition ( f
ac
~1/τ
sh
).
In such case, one may use longer DNA tethers to increase t
c, x
in
respect of τ
sh
and extract a calibration table for F
mag
as a function of
the magnets distance to the magnetic bead [59]. However, this
only works for magnetic beads with a small dispersion in magnetic
content, hence in force, such as the Dynabeads M-270
(R = 1.4 μm) and MyOne (R = 0.5 μm) magnetic beads from

Invitrogen. For shorter tethers or higher forces, one may use a very
fast camera, i.e., f
ac
in the kilohertz range, or use a nonzero dead
time acquisition, i.e., τ
sh
≪ 1/f
ac
[59, 61]. The former is not
available for all camera models and only when using a small field
of view [28–30].The latter is easily programmable in most cameras
without compromising the field of view [61]. Another possibility is
to correct hδx
2
i
meas
for the camera image blurring, either in the
frequency or time domain [59, 64]. This works well for τ
sh
/2 < t
c,
x
< τ
sh
/4 [60], and packages in MATLAB and Python are available
to perform such calibrations [59, 65]. These strategies however fail
to perform accurate force calibration for very short tethers, as the
rotation of the bead induced by the magnetic field pinning by the
magnetic bead must be accounted for [66, 67]. Similar strategies to
calibrate the force may also be applied to acoustic force spectros-
copy (AFS) [68], as the tethered bead is described by a similar
model (i.e., the inverted pendulum).
An Introduction to Magnetic Tweezers 385
3.3 Estimating the
Spatiotemporal
Resolution of Magnetic
Tweezers
The main parameters measured in magnetic tweezers experiments
are the change in the tether’s extension L
ext
due to either a mechan-
ical response of the tether or an enzymatic activity modifying the
tether length. It is therefore essential to determine the noise ampli-
tude along the z-axis. The spatiotemporal resolution in a magnetic
tweezers assay depends on the tracking and thermal noise as
follows:
δztot
hi
= δztr 2 þ δzth 2:ð13Þ
3.3.1 Tracking
Resolution and Stability
The tracking resolution is defined by the hardware (microscope
objective magnification, numerical aperture, pixel size and light
intensity) and the algorithm used. To experimentally evaluate δz
tr
,
the Allan deviation (AD) is particularly useful [28, 64, 69]
(Fig. 5a). The AD of a particle position along the, e.g., z-axis, is
defined as follows:
σAD τðÞ= 1
2 zτ,jþ1-zτ,j
2 with zτ,j = 1
τ
τ jþ0:5ðÞ
τ j-0:5ðÞ
ztðÞdt, ð14Þ
where τ defines both the time between consecutive samples and the
time over which the sample is averaged. Simply put, the AD is
one-half the average difference in position between consecutive
intervals of duration τ, averaged over all intervals of duration τ.
For a bead stuck to the surface, we observe two regimes (Fig. 5a):
AD initially decreases as 1= τ
p , indicating how the frame-to-frame
uncorrelated noise averages out, and the AD subsequently reaches a
lower bound and rises again due to long timescale drift dominating
the noise (e.g., mechanical drift, tracking algorithm bias). To
improve the stability during the measurement, the mechanical
drift is corrected by subtracting the position of a reference bead

fixed to the flow chamber surface from the position of the magnetic
bead (Figs. and )[ ]. The resolution of the bead position
tracking may be further improved by setting an autofocus locked
onto the position of a reference bead and adjusting the objective’s
285a1a
386 David Dulin
Fig. 5 Spatiotemporal resolution of a magnetic tweezers instrument. (a) Allan deviation (AD) of the z-axis
position of a 3 μm diameter surface-attached polystyrene reference bead (blue), subtracted to another
reference bead (RS, orange), and using autofocus (AF, green). The data were acquired using a 100× objective
magnification and at 58 Hz acquisition frequency. (b) Raw (gray) and 1 Hz low -pass filtered (dark gray) trace
acquired while using the autofocus and drift corrected by subtracting the z-position of another reference bead
(green in (a)) (Adapted from Ref. [49]). (c, d) Height of a (c) reference bead and (d) DNA-tethered bead while
using the piezo stage to move the sample by the increments indicated on top of the panel (in nm) (Adapted
from Ref. [69]). (e, f) Magnetic tweezers assay to monitor single-nucleotide steps of Upf1 helicase when
unwinding a DNA hairpin (Adapted from Ref. [39])

ÞÞ
focal plane’s position using a high-resolution piezo stage and
increase the τ at which AD rises again by decreasing the negative
impact of the tracking algorithm bias [49] (Fig. 5a, b). For a
magnetic tweezers instrument with 100× magnification, 1.25
numerical aperture microscope objective, and 60 nm pixel size in
the image plane, tracking resolutions for a single image of δz
tr
~1 nm
and δz
tr
~0.3 nm are achievable for 1 μm and 3 μm diameter beads,
respectively (using the quadrant interpolation algorithm) [69].
An Introduction to Magnetic Tweezers 387
The tracking resolution may be improved by acquiring data at
high f
ac
and subsequently averaging out the tracking noise by
integrating the bead position over N frames. This results in a
reduction of the tracking noise by a factor of N
p , enabling the
observation of steps as small as 0.3 nm for a reference bead with the
standard magnetic tweezers configuration (Fig. 5c)[28–30, 69]
and for a tethered magnetic bead (Fig. 5d)[69]. Recent develop-
ments in magnetic tweezers instrumentation, specifically in the
illumination and imaging path, have enabled the first observation
of single-nucleotide translocation steps by a helicase unwinding a
DNA hairpin (Fig. 5e, f)[39].
3.3.2 Thermal Noise The thermal noise depends on the tether stiffness, k
z
, as follows:
δz2
th =kBT=kz,ð15Þ
where
kz =
∂Fmag Lext
ðÞ
∂Lext
= kBT
2LPLC 2þ 1-Lext
LC ð16Þ
with L
P
and L
C
being the persistence and the contour length of the
tether, respectively, assuming the response of the tether to F
mag
is
well-described by the inextensible Worm-like chain model
[70]. Similar to hδx
2
i
meas
, the thermal noise is integrated by the
camera during the image acquisition. Hence,
δz2
th meas = 2kBT=πkz
ðÞ arctan 4πtc,z=τsh ð17Þ
with t
c, z
= γ/k
z
. For L
ext
≤ R, γ must be corrected to account for
the coverslip surface effect using Brenner’s approximation [63]:
γBrenner =6πηR= 1-9=8 R= Rþ Lext
ðÞðÞþ 3=8 R= Rþ Lext
ðð3
-1=4 R= Rþ Lext
ðÞðÞ
4 ,
ð18Þ
Ideally, the resolution of the magnetic tweezers assay is limited
by the thermal noise, which can be estimated using Eqs. 16 and 17.
An accurate simulation of the overall measurement noise for a
tethered magnetic bead in a magnetic tweezers assay has been
described by Burnham and colleagues [71], which is useful to

estimate the spatiotemporal resolution for a given magnetic twee-
zers experiment.
388 David Dulin
Fig. 6 DNA supercoiling experiments using magnetic tweezers. Dynamic
rotation-extension experiment on a 21 kbp long DNA tether at either 0.3 pN
(gray) or 4 pN (black). At low force, both negative and positive supercoils induce
plectonemes. At high force, positive supercoils induce plectonemes, while
negative supercoils unwind the DNA tether. The arrow indicates the buckling
transition at high force
3.4 Using Torque
Spectroscopy in
Magnetic Tweezers
One of the key aspects of magnetic tweezers is their ability to
control the torque applied to a coilable biomolecule (Fig. 1d,
Fig. 6)[6, 57]. This has been (and still is) used to investigate the
mechanical response to torque of double-stranded nucleic acids
[15, 72]. To enable torque spectroscopy, the nucleic acid must be
topologically constrained, i.e., without free rotation point, such as a
fully double-stranded DNA with multiple attachment points at
both ends (Fig. 1d). The twist (Tw) and the writhe (Wr) define
the supercoiled state of the molecule. The former is the number of
times the molecule turns around itself, such as for the DNA double
helix, and the latter is defined by the number of times the molecule
winds over itself. The helical pitch for a relaxed DNA molecule is
10.5 bp/turn and, therefore, the total twist in a relaxed DNA
molecule (Tw
0
) is the number of base pairs divided by the helical
pitch. The linking number (Lk), which is the sum of twist and
writhe, is a topological invariant for a torsionally constrained mole-
cule, meaning
Lk =Tw þ Wr =constant:ð19Þ
For a torsionally relaxed DNA molecule, Wr = 0, so Lk
0
= Tw
0
.
A molecule is said to be supercoiled when Lk ≠ Lk
0
.
In addition, the supercoil density σ is a useful description of the
torsional state of a molecule:

An Introduction to Magnetic Tweezers 389
σ = Lk -Lk0
ðÞ=Lk0:ð20Þ
L
ext
as a function of σ is often used to represent magnetic
tweezers experiments investigating the response of double-
stranded nucleic acids to torsional stress. This provides an easy
way to compare the torsional properties of DNA tethers of different
lengths.
Upon addition of positive turns to a torsionally relaxed mole-
cule, L
ext
remains constant at first, as the addition of twist is
absorbed through deformation of the molecule (Fig. 6). In this
regime, Tw > Tw
0
,Wr = 0, and the torque Γ increase linearly with
the number of turns N:
Γ=C2πN=LC,ð21Þ
where C is the torsional modulus of the molecule, e.g., C~90 k
B
T
for DNA [35, 73–75]. This may be used to monitor the torque-
dependence of a specific DNA-protein interaction, e.g., RNA
polymerase-promoter open complex formation by the bacterial
RNA polymerase [76]. At the critical torque Γ
C
, the molecule’s
extension suddenly decreases to form the first loop upon further
addition of coiling to the DNA molecule (Fig. 6). This event is also
called the buckling transition and is followed by a linear decrease in
L
ext
with added turns [6, 77]. Γ
C
is given through having the
energy to form a loop of radius R
L
being equal to the work done
by the addition of one extra turn 2πΓ
C
:
ER =2πRLFmag þ πLPkBT=RL,ð22Þ
Minimizing E
R
as a function of R
L
gives Γ
C
and the change in
extension per superhelical turn Δz such as [73, 78]:
ΓC = 2LPkBTFmag andΔz =2πRL =π 2LPkBT=Fmag, ð23Þ
This model only describes Δz qualitatively, and more sophisti-
cated models have been derived to describe Δz more accurately
[15, 31]. At high force and in the negative supercoil regime, it is
more favorable for the DNA molecule to unwind than forming
plectoneme, while the rotation-extension of a DNA molecule is
symmetrical at low force, i.e., the tether forms plectonemes for
both negative and positive supercoil addition (Fig. 6).
4 Combining Magnetic Tweezers with Other Techniques
Magnetic tweezers have been combined with fluorescence micros-
copy to enable simultaneous force/torque and fluorescence spec-
troscopy investigations. The preferred fluorescence approach is
objective-based total internal reflection fluorescence microscopy
(TIRFM), as it is a surface-based approach with a shallow excitation
depth (~hundreds of nanometers), leaving the magnetic bead out

of the excitation volume. Two main configurations have been
reported for such assay, using either a standard vertical magnet
configuration (Fig. 7a) or a horizontal magnet configuration
(Fig. 7b). In the former configuration, the magnets pull vertically
on the magnetic bead, and vertical motion of a dye-labeled enzyme
may be reported using the fluorescence channel and the exponen-
tial decay of the evanescent field of the TIRF excitation [79–82]. In
the latter configuration, a magnetic force is applied sideways, which
stretches the DNA molecule laterally, enabling transverse observa-
tion of displacements biomolecular objects (e.g., protein, plecto-
neme) [83–85]. These two configurations have different
advantages. The first enables rapid modulation of the applied
torque, as is done in the rotor bead assay developed by Bryant
and colleagues [12]. A sideway configuration gives access to a
higher localization precision of fluorescently labeled biomolecules
moving along, e.g., a DNA tether [83–85]. Chapter 22 will discuss
different configurations of magnetic tweezers combined with fluo-
rescence microscopy. Darkfield microscopy has been combined
with magnetic tweezers in the rotor bead assay to use backscattered
light from a gold nanoparticle as a tracker, which provides an
excellent signal-to-noise ratio while minimizing the size of the
object to track, and therefore gives access to a higher measurement
bandwidth [86]. The combination of magnetic tweezers with opti-
cal tweezers has also been reported by Cees Dekker and
colleagues [87].
390 David Dulin
Fig. 7 Magnetic tweezers combined with total internal reflection fluorescence (TIRF) microscop y. (a) Vertically
oriented attractive force and (b) horizontally oriented attractive force. The pink oval indicates a protein of
interest labeled with a fluorescent dye (green sphere). In (a), the exponential decay of the evanescent field is
used to localize the protein along the DNA, while in (b) the position is determined from direct localization in the
plane of observation

An Introduction to Magnetic Tweezers 391
5 Applications of Magnetic Tweezers in Single-Molecule Biophysics
Magnetic tweezers present many advantages to study biomolecules
in vitro one at a time. They are a force clamp technique that enables
force spectroscopy measurements from ~10 fN to ~1 nN [4],
depending on the total amount of magnetic material in the bead.
Because the distance between the magnetic bead and the magnets
has to vary significantly (around 0.05 mm) to vary the force signifi-
cantly, magnetic tweezers can apply a constant force over very long
measurement. This holds true even at low force (< 1 pN), unlike
for an AFM or optical tweezers. Furthermore, the combination of a
homogenous magnetic field over a very large field of view (~mm
2
)
and commercially available magnetic beads with homogenous mag-
netic content enables high-throughput force spectroscopy mea-
surements at constant force with a small bead-to-bead variation in
force (~10% standard deviation). For these reasons, magnetic twee-
zers have been applied to study protein folding and unfolding
dynamics at a low constant force [19], such as the titin immuno-
globulin domain [23], talin and protein L [21], von Willebrand
factor folding [20], protein-ligand interactions to interrogate
either SARS-CoV-2 spike or ACE2 interactions (Fig. 8a, b)[88],
and the rapamycin-mediated association between FKBP12 and
FRB [22, 89].
Furthermore, the recent advances in tracking algorithms, illu-
mination, and imaging strategies have brought magnetic tweezers
on par with optical tweezers in terms of their spatiotemporal reso-
lution. Additionally, the ability to perform high-throughput track-
ing in magnetic tweezers enables the in-depth characterization of
mechanochemical pathways of translocating molecular motor.
Examples include viral RNA polymerases (Fig. 8c, d)[26, 49,
90–92], the bacterial RNA polymerase [93], the DNA polymerase
[7, 8], helicases (Fig. 5c)[39, 44, 94–104], and the SMC complex
[105, 106]. Magnetic tweezers have also enabled the characteriza-
tion of nucleoprotein filament formation or mechanical properties
[107–110], protein-mediated DNA condensation [111, 112], and
chromatin filament and nucleosome stability [113–117].
Magnetic tweezers are naturally well suited to perform torque
spectroscopy experiments. This has been extensively used to inves-
tigate biological systems that induce a change in the linking number
of a tethered coilable double-stranded nucleic acid. Topoisome-
rases remove the excess of negative or positive supercoils in the
DNA molecule that naturally occur in the cell during DNA tran-
scription and replication [118]. Therefore, their activity is essential
to maintain cellular homeostasis. Using magnetic tweezers to
induce a large excess of supercoils to a DNA molecule has been
used to investigate the mechanochemical cycle of topoisomerases
[24, 119–124]. Cellular RNA polymerases have been extensively

392 David Dulin
Fig. 8 Examples of magnetic tweezers applications in single-molecule biophysics. (a, b) Schematic and
experimental traces showing the binding and dissociation kinetics of the SARS-CoV-2 spike protein RBD from
the ACE2 receptor as a function of force. (b) The time-dependent traces reveal populations in the bound and
dissociated states as a function of the applied force (Adapted from Ref. [88]). (c, d) Schematic and
experimental traces of elongating SARS-CoV-2 core replication-transcription complexes. (d) The time-
dependent traces demonstrate rich dynamics with bursts of nucleotide addition interrupted by pauses of
various durations (Adapted from Ref. [49]). (e, f) RNA polymerase (RNAP)-promoter open complex formation on
a positively supercoiled DNA. Upon promoter opening, upon n positive supercoils addition, moving the bead
downward by nΔz. The surface is passivated using a lipid-bilayer strategy. The trace in (f) shows the promoter
alternating between a closed state (CS, promoter closed) and an open state (OS, RNAP-promoter open)
(Adapted from Ref. [50])

investigated using magnetic tweezers. Indeed, they must open
double-stranded DNA at the promoter site to form the RNA
polymerase-promoter open complex and initiate transcription,
which consequently removes ~1 turn of twist in the DNA molecule
(the transcription bubble is 13–14 nt, and DNA makes one full turn
every ~10.5 bp). Using the conservation of the linking number
condition described above (Fig. 8e), many details in the mechanism
of transcription initiation by cellular RNA polymerase have been
revealed (Fig. 8f), such as the impact of torque on promoter open-
ing, the dynamics of transcription initiation and promoter escape as
a function of the promoter sequence and salt concentration, the
transcription start site selection, and R-loop formation during
transcription [14, 50, 76, 125–129]. A similar approach was used
to investigate Cas9 R-loop formation [130, 131]. The torque
spectroscopy capabilities of magnetic tweezers have also been
used to investigate the torsional properties and stability of chroma-
tin filaments [116, 132, 133] and nucleosome assembly [134–
136], as well as other nucleoprotein filaments, such as those formed
by Rad51, RecA, and H-NS [35, 137–141].
An Introduction to Magnetic Tweezers 393
6 Perspectives
The unique advantages of magnetic tweezers, i.e., simplicity (and
therefore low cost), stability, high parallelization and resolution,
large force range, and torque spectroscopy, make it a very powerful
technique. It has been established in many labs worldwide, with an
ever-increasing demand. Only one company currently sells mag-
netic tweezers instruments (Mad City labs), and more could be
done to have the technique more available at low cost. Currently,
no open-source instrument design has been released or published,
and CAD drawings would help democratizing magnetic tweezers.
A software interface already exists [25], though in proprietary
format (LabView, National Instruments), and efforts must be
made to release an open-source interface in a nonproprietary lan-
guage, such as Python. The development of routines for data
analysis in nonproprietary languages to help the analysis of complex
dynamics of molecular motors will further support the democrati-
zation of magnetic tweezers with appropriate statistical tools to
analyze single-molecule data. Altogether, these developments will
bring magnetic tweezers and their application to a broader com-
munity. Lastly, there is no combined high-throughput single-
molecule force/torque and fluorescence spectroscopy assay avail-
able to date. The statistical power of such hydrid assay would
potentiate the investigation of ever more complex biomolecular
systems.

394 David Dulin
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