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.

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 cellular mechanics is usually relatively large (≫ 1 nN) [2, 3] compared 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.

In the 1990s, the Bustamante lab and the Croquette and Bensimon lab pioneered the modern version of single-molecule 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 polymerases [7,8,9], topoisomerases and gyrases [10,11,12,13], cellular and viral RNA polymerases [14], nucleoprotein filaments [15,16,17,18], and the mechanical stability of protein folding and protein-ligand interactions [19,20,21,22,23]. 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 biomolecule (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 complementary metal-oxide-semiconductor (CMOS) cameras enable both high-throughput (Fig. 1b) [24,25,26,27] and high-speed measurements [28,29,30]. 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.

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])

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 permanent magnets, being either vertically (Fig. 1a, c, and d) or horizontally 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 additionally 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].

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 collimated using a high numerical aperture aspherical lens, and provide enough light intensity to image the beads in standard image acquisition 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 requirement. 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 collimation 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 illumination, i.e., by blocking the zero-order light pathway, has recently demonstrated the best to date resolution by reducing the background 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 developed, such as the quadrant interpolation method [25, 42]. To determine 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].

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 diffraction 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

2.4 Temperature Control

Enzymatic reactions are sensitive to temperature fluctuations and follow the Arrhenius law:

$$ k(T)=A{e}^{-{E}_{\mathrm{a}}/{k}_{\mathrm{B}}T}, $$

(1)

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 [44,45,46]. 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 reaction (±0.1 °C), which can be achieved using a simple device commercially available from Thorlabs [46].

2.5 Surface Functionalization and Nucleic Acid Construct Fabrication

Magnetic tweezers are a surface-based technique (Fig. 1a); therefore, the flow chamber’s glass surface should be treated with care to prevent nonspecific attachment to the surface of either the magnetic beads or the biomolecules of interest. Different types of surface functionalization have been developed, such as polyethylene 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 streptavidin 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 nitrocellulose 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 biomolecule 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 [51,52,53,54,55,56], and this topic will therefore not be further discussed here.

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 systems. The force experienced by a magnetic particle in a magnetic field is described by:

$$ {\overrightarrow{F}}_{\mathrm{mag}}=\frac{1}{2}\overrightarrow{\nabla}\left({\overrightarrow{m}}_{\mathrm{sat}}\cdot \overrightarrow{B}\right), $$

(2)

where ${\overrightarrow{F}}_{\mathrm{mag}}$ is the magnetic force, ${\overrightarrow{m}}_{\mathrm{sat}}$ is the saturated magnetization of the particle, and $\overrightarrow{B}$ is the magnetic field [32]. Interestingly, 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 ${\overrightarrow{m}}_{\mathrm{sat}}$, with a minor component ${\overrightarrow{m}}_0$ not aligned with $\overrightarrow{B}$ [58]. This induces a torque $\overrightarrow{\Gamma}$ on the bead, which is derived from:

$$ \overrightarrow{\Gamma}={\overrightarrow{m}}_0\times \overrightarrow{B}. $$

(3)

The torque response of the biomolecule is negligible in respect to $\overrightarrow{\Gamma}$. Hence, rotating the magnets induces a rotation of the magnetic 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 magnitude of $\overrightarrow{\Gamma}$, 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 characterizing 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 $\overrightarrow{B}$ (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 constrained by the magnetic field (Fig. 4a), and the length of the pendulum is therefore L_ext + R, where R is the magnetic bead radius.

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])

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:

$$ {F}_{\mathrm{restoring}}={k}_x\delta x, $$

(4)

where k_x is the trap stiffness along the x-axis. Therefore,

$$ {F}_{\mathrm{restoring}}={F}_{\mathrm{mag}}\cdot \sin \left(\theta \right)={F}_{\mathrm{mag}}\cdot \delta x/{L}_{\mathrm{ext}}, $$

(5)

and

$$ {k}_x={F}_{\mathrm{mag}}/{L}_{\mathrm{ext}}, $$

(6)

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.,

$$ {U}_{\mathrm{x}}=\left(1/2\right)\cdot {k}_{\mathrm{x}}{\delta x}^2, $$

(7)

and we can therefore apply the equipartition theorem (〈U_x〉 = (1/2) ∙ k_BT) on Eq. (7), and we obtain:

$$ \left\langle {\delta x}^2\right\rangle ={k}_{\mathrm{B}}T/{k}_{\mathrm{x}}={k}_{\mathrm{B}}T\cdot {L}_{\mathrm{ext}}/{F}_{\mathrm{mag}}, $$

(8)

and, by extension, for the long pendulum case:

$$ \left\langle {\delta y}^2\right\rangle ={k}_{\mathrm{B}}T/{k}_{\mathrm{y}}={k}_{\mathrm{B}}T\cdot \left({L}_{\mathrm{ext}}+R\right)/{F}_{\mathrm{mag}}. $$

(9)

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 magnetic 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:

$$ {t}_{\mathrm{c},\mathrm{x}}=\gamma /{k}_{\mathrm{x}}=6\pi \eta R{L}_{\mathrm{ext}}/{F}_{\mathrm{mag}}\ \mathrm{and}\ {t}_{\mathrm{c},\mathrm{y}}=\gamma /{k}_{\mathrm{y}}=6\pi \eta R\left({L}_{\mathrm{ext}}+R\right)/{F}_{\mathrm{mag}}, $$

(10)

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 Faxén law [63]:

$$ {\gamma}_{\mathrm{Fax}}\acute{e}{\mathrm{n}}=6\pi \eta R/\left[\begin{array}{lll}1-9/8\left(R/\left(R+{L}_{\mathrm{e}\mathrm{xt}}\right)\right)+1/2{\left(R/\left(R+{L}_{\mathrm{e}\mathrm{xt}}\right)\right)}^3-57/100{\left(R/\left(R+{L}_{\mathrm{e}\mathrm{xt}}\right)\right)}^4\\ {}+1/5{\left(R/\left(R+{L}_{\mathrm{e}\mathrm{xt}}\right)\right)}^5+7/200{\left(R/\left(R+{L}_{\mathrm{e}\mathrm{xt}}\right)\right)}^{11}-1/25{\left(R/\left(R+{L}_{\mathrm{e}\mathrm{xt}}\right)\right)}^{12}\end{array}\right], $$

(11)

〈δx²〉 averages away toward a measured value 〈δx²〉_meas as a function of the camera shutter time τ_sh and t_{c, x} as

$$ {\left\langle {\delta x}^2\right\rangle}_{\mathrm{meas}}=\left(2{k}_{\mathrm{B}}T/\pi {k}_{\mathrm{x}}\right)\arctan \left(4\pi {t}_{\mathrm{c},\mathrm{x}}/{\tau}_{\mathrm{sh}}\right). $$

(12)

To minimize the error in the force due to camera image blurring, we must minimize the difference between 〈δx²〉 and 〈δx²〉_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 [28,29,30].The latter is easily programmable in most cameras without compromising the field of view [61]. Another possibility is to correct 〈δx²〉_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 spectroscopy (AFS) [68], as the tethered bead is described by a similar model (i.e., the inverted pendulum).

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 mechanical response of the tether or an enzymatic activity modifying the tether length. It is therefore essential to determine the noise amplitude along the z-axis. The spatiotemporal resolution in a magnetic tweezers assay depends on the tracking and thermal noise as follows:

$$ \left\langle {\delta z}_{\mathrm{tot}}\right\rangle =\sqrt{\delta {z_{\mathrm{tr}}}^2+\delta {z_{\mathrm{th}}}^2}. $$

(13)

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:

$$ {\sigma}_{\mathrm{AD}}\left(\tau \right)=\sqrt{\frac{1}{2}\left\langle {\left({\overline{z}}_{\uptau, \mathrm{j}+1}-{\overline{z}}_{\uptau, \mathrm{j}}\right)}^2\right\rangle }\ \mathrm{with}\ {\overline{z}}_{\uptau, \mathrm{j}}=\frac{1}{\tau }{\int}_{\tau \left(j-0.5\right)}^{\tau \left(j+0.5\right)}z(t) dt, $$

(14)

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 $1/\sqrt{\tau }$, 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. 1a and 5a) [28]. 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 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].

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])

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$\sqrt{N}$, enabling the observation of steps as small as 0.3 nm for a reference bead with the standard magnetic tweezers configuration (Fig. 5c) [28,29,30, 69] and for a tethered magnetic bead (Fig. 5d) [69]. Recent developments 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:

$$ \left\langle \delta {z}_{\mathrm{th}}^2\right\rangle ={k}_{\mathrm{B}}T/{k}_{\mathrm{z}}, $$

(15)

where

$$ {k}_{\mathrm{z}}=\frac{\partial {F}_{\mathrm{mag}}\left({L}_{\mathrm{ext}}\right)}{\partial {L}_{\mathrm{ext}}}=\frac{k_{\mathrm{B}}T}{2{L}_{\mathrm{P}}{L}_{\mathrm{C}}}\left(2+\left(1-\frac{L_{\mathrm{ext}}}{L_{\mathrm{C}}}\right)\right) $$

(16)

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 〈δx²〉_meas, the thermal noise is integrated by the camera during the image acquisition. Hence,

$$ {\left\langle \delta {z}_{\mathrm{th}}^2\right\rangle}_{\mathrm{meas}}=\left(2{k}_{\mathrm{B}}T/\pi {k}_{\mathrm{z}}\right)\ \arctan \left(4\pi {t}_{\mathrm{c},\mathrm{z}}/{\tau}_{\mathrm{sh}}\right) $$

(17)

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]:

$$ {\displaystyle \begin{array}{l}{\gamma}_{\mathrm{Brenner}}=6\pi \eta R/\left[1-9/8\left(R/\left(R+{L}_{\mathrm{ext}}\right)\right)+3/8{\left(R/\left(R+{L}_{\mathrm{ext}}\right)\right)}^3\right.\\ {}\left.-1/4{\left(R/\left(R+{L}_{\mathrm{ext}}\right)\right)}^4\right],\end{array}} $$

(18)

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 tweezers experiment.

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 (Tw₀) 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 molecule, meaning

$$ \mathrm{Lk}=\mathrm{Tw}+\mathrm{Wr}=\mathrm{constant}. $$

(19)

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

For a torsionally relaxed DNA molecule, Wr = 0, so Lk₀ = Tw₀. A molecule is said to be supercoiled when Lk ≠ Lk₀.

In addition, the supercoil density σ is a useful description of the torsional state of a molecule:

$$ \sigma =\left(\mathrm{Lk}-{\mathrm{Lk}}_0\right)/{\mathrm{Lk}}_0. $$

(20)

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 molecule, L_ext remains constant at first, as the addition of twist is absorbed through deformation of the molecule (Fig. 6). In this regime, Tw > Tw₀, Wr = 0, and the torque Γ increase linearly with the number of turns N:

$$ \Gamma =C2\pi N/{L}_{\mathrm{C}}, $$

(21)

where C is the torsional modulus of the molecule, e.g., C~90 k_BT for DNA [35, 73,74,75]. 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:

$$ {E}_{\mathrm{R}}=2\pi {R}_{\mathrm{L}}{F}_{\mathrm{mag}}+\pi {L}_{\mathrm{P}}{k}_{\mathrm{B}}T/{R}_{\mathrm{L}}, $$

(22)

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]:

$$ {\Gamma}_{\mathrm{C}}=\sqrt{2{L}_{\mathrm{P}}{k}_{\mathrm{B}}T{F}_{\mathrm{mag}}}\ \mathrm{and}\Delta z=2\pi {R}_{\mathrm{L}}=\pi \sqrt{2{L}_{\mathrm{P}}{k}_{\mathrm{B}}T/{F}_{\mathrm{mag}}}, $$

(23)

This model only describes Δz qualitatively, and more sophisticated 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 microscopy to enable simultaneous force/torque and fluorescence spectroscopy 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 exponential decay of the evanescent field of the TIRF excitation [79,80,81,82]. In the latter configuration, a magnetic force is applied sideways, which stretches the DNA molecule laterally, enabling transverse observation of displacements biomolecular objects (e.g., protein, plectoneme) [83,84,85]. 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 [83,84,85]. Chapter 22 will discuss different configurations of magnetic tweezers combined with fluorescence 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 optical tweezers has also been reported by Cees Dekker and colleagues [87].

Fig. 7

Magnetic tweezers combined with total internal reflection fluorescence (TIRF) microscopy. (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

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 significantly, 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 (~mm²) and commercially available magnetic beads with homogenous magnetic content enables high-throughput force spectroscopy measurements at constant force with a small bead-to-bead variation in force (~10% standard deviation). For these reasons, magnetic tweezers have been applied to study protein folding and unfolding dynamics at a low constant force [19], such as the titin immunoglobulin 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].

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])

Furthermore, the recent advances in tracking algorithms, illumination, and imaging strategies have brought magnetic tweezers on par with optical tweezers in terms of their spatiotemporal resolution. Additionally, the ability to perform high-throughput tracking 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, 90,91,92], the bacterial RNA polymerase [93], the DNA polymerase [7, 8], helicases (Fig. 5c) [39, 44, 94,95,96,97,98,99,100,101,102,103,104], and the SMC complex [105, 106]. Magnetic tweezers have also enabled the characterization of nucleoprotein filament formation or mechanical properties [107,108,109,110], protein-mediated DNA condensation [111, 112], and chromatin filament and nucleosome stability [113,114,115,116,117].

Magnetic tweezers are naturally well suited to perform torque spectroscopy experiments. This has been extensively used to investigate biological systems that induce a change in the linking number of a tethered coilable double-stranded nucleic acid. Topoisomerases remove the excess of negative or positive supercoils in the DNA molecule that naturally occur in the cell during DNA transcription 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, 119,120,121,122,123,124]. Cellular RNA polymerases have been extensively 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 opening, 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, 125,126,127,128,129]. 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 chromatin filaments [116, 132, 133] and nucleosome assembly [134,135,136], as well as other nucleoprotein filaments, such as those formed by Rad51, RecA, and H-NS [35, 137,138,139,140,141].

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 magnetic 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 language, 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 democratization 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 community. Lastly, there is no combined high-throughput single-molecule force/torque and fluorescence spectroscopy assay available to date. The statistical power of such hydrid assay would potentiate the investigation of ever more complex biomolecular systems.