Force spectroscopy with electromagnetic tweezers

Abstract

Force spectroscopy using magnetic tweezers (MTs) is a powerful method to probe the physical characteristics of single polymers. Typically, molecules are functionalized for specific attachment to a glass surface at one end and a micrometer-scale paramagnetic bead at the other end. By applying an external magnetic field, multiple molecules can be stretched and twisted simultaneously without exposure to potentially damaging radiation. The majority of MTs utilize mobile, permanent magnets to produce forces on the beads (and the molecule under test). However, translating and rotating the permanent magnets may require expensive precision actuators, limit the rate at which force can be changed, and may induce vibrations that disturb tether dynamics and bead tracking. Alternatively, the magnetic field can be produced with an electromagnet, which allows fast force modulation and eliminates motor-associated vibration. Here, we describe a low-cost quadrapolar electromagnetic tweezer design capable of manipulating DNA-tethered MyOne paramagnetic beads with forces as high as 15 pN. The solid-state nature of the generated B-field modulated along two axes is convenient for accessing the range of forces and torques relevant for studying the activity of DNA motor enzymes like polymerases and helicases. Our design specifically leverages technology available at an increasing number of university maker spaces and student-run machine shops. Thus, it is an accessible tool for undergraduate education that is applicable to a wide range of biophysical research questions.

I. INTRODUCTION

A. Magnetic tweezers in biophysical research

Magnetic tweezers (MTs) comprise a class of force spectroscopy tools that are well-suited to the characterization and manipulation of individual biomolecules, polymers, and even live cells.¹ There are several methods with which to exert forces on microscale particles and manipulate attached polymers, but magnetic tweezers are perhaps the most straightforward and comparatively inexpensive to implement. In addition, the trapped particles can be rotated to exert torsion on attached polymers, which is not trivial with other methods. The utility of these instruments can be well demonstrated in studies of DNA, a well-characterized and experimentally convenient polymer with broad interest and applications in biophysical research. Magnetic manipulators were first presented by Strick et al.² to study the elasticity of double-stranded DNA. Since then, investigators have used MT to experiment on modified and unmodified nucleic acid polymers in a variety of buffers.^1,3–7 DNA is a particularly convenient sample due to the range of enzymatic tools that can be used to prepare large quantities of identical polymers with specific lengths and sequences. In the aqueous buffer environment, DNA conformations can be modified by small molecules and salts^8–10 as well as through interactions with processive enzymes^11,12 and other proteins.^13,14 Generally, MTs require that the molecule under test be functionalized for specific attachment to a fixed glass surface at one end and a micrometer-scale paramagnetic bead (PMB) at the other end. The Brownian motion of the tethered bead is confined by the DNA tether anchoring it to a point on the glass. Monitoring the position of the bead as a function of time reveals the anchor point and the effective length of the tether connecting the bead. When an external magnetic field is applied to the PMB, the bead experiences a force proportional to the gradient of the field.¹⁵ Furthermore, if the magnetic moment of the bead is not aligned with the field, the PMB will experience a torque. Thus, as the PMB responds to the external field, the tethering molecule may stretch and twist [see Fig. 1(a)]. By tracking the motion of the bead with a microscope, the mechanical properties of the tethering molecule can be inferred.

FIG. 1.

Magnetic bead-field interactions. (a) A molecule of DNA is shown immobilized to a glass surface (blue) and attached to a magnetic bead (purple). The magnetic moment of the bead (yellow) aligns itself with the external magnetic field (B, green) propagating in the +z direction from a magnet above (not shown). Left: The gradient of the B field attracts the bead along z (red). Right: A top-down view. Torque is generated when the magnetic field (green) rotates about a normal to the xy-plane. If there is no swivel in the attachment of the DNA to the bead, the DNA tether is twisted causing supercoiling. (b) The conformation of DNA depends on tension and torque. An untwisted molecule under low tension (a) extends as the tension increases (b). Alternatively, twisting the DNA under low tension supercoils the molecule into plectonemic loops, reducing the tether length (c). Raising the tension on the highly supercoiled molecule converts the writhe back to twist as the DNA loops are stretched (d).

A significant feature of magnetic tweezing is the ability to arbitrarily manipulate DNA topology in real time. This is especially relevant because DNA is normally topologically constrained in vivo via tension and twist, conditions that are exceedingly difficult to regulate biochemically in vitro.^16,17 In response to a mild attractive force that draws the bead away from the anchor point, the DNA tether stretches like a spring, exerting entropic force. Torque applied to rotate the bead twists the attached molecule, imparting supercoiling. When supercoiling exceeds a critical threshold, the molecule buckles and writhes into a plectonemic form, shortening the overall tether length. Thus, the effective tether length changes in response to topological manipulations of DNA via tension and twist [Fig. 1(b)]. The breakthrough associated with MT in biophysical research stems from the fact that DNA processing enzymes like polymerases impart tension and twist within particular force and torque regimes. Using this tool, researchers have investigated not only DNA mechanics but also the activity of DNA topoisomerases,¹¹ helicases,¹⁸ and RNA polymerases,¹⁹ contributing to our understanding of their respective mechanisms.

B. Permanent, rare-earth magnetic tweezers

Most magnetic tweezers utilize a pair of permanent, rare-earth magnets near the sample. Field strength, as well as gradient, at the sample plane is modulated by mechanically moving the magnets closer to or further from the sample. Similarly, field orientation is controlled by rotating the magnets. Although this design generates the necessary magnetic forces, it has a few drawbacks. The motors may induce vibrations that compromise bead tracking and limit the accuracy and resolution with which tether length and force can be measured. Additionally, the sample is usually illuminated by a beam passing through a narrow gap between the magnets. Thus, the magnets must be carefully aligned with the objective to avoid variations in illumination strength as the magnets rotate or translate. Lastly, the magnetic field may be axially asymmetric, causing a tethered bead to precess as the magnets are turned. To ameliorate these effects, data from permanent magnetic tweezers are often recorded only at integral turns of the magnetic field that twists the tether.

II. ELECTROMAGNETIC TWEEZERS

These limitations associated with permanent magnets may be overcome by replacement with an electromagnet. Electromagnetic tweezers (eMTs) use a set of solenoids to generate the magnetic field. The field strength and orientation is modulated by changing the current through the solenoids. Such modulation can be more rapid than that achieved by moving a permanent magnet to alter the magnetic field strength and orientation. Previous implementations of electromagnetic tweezers have been reported. A hexagonal pattern of solenoids with Mu-metal cores was described which produced forces on super-paramagnetic beads as high as 20 pN vertically and 5 pN horizontally and rotations at 10 Hz. The large potential well of this system even allowed horizontal positioning of 5  $\mu$m-diameter magnetic beads.⁴ In another implementation, the recording head of an audio tape device was employed to assemble a rotationally static tweezer that produced up to 50 pN with 1 A of current and allowed high bandwidth (10 kHz) force modulation.²⁰ A tweezer with four laterally set solenoids generating a horizontal field for rotation and another axially placed solenoid adding a vertical component generated fields as high as 8 pN with 3 A of current.²¹ A similar instrument with separate rotational and axial control was formed of Helmholtz coils that create rotational stiffness of several $\mathrm{pN}\mu$m/rad and an axially placed permanent magnet to apply tension up to tens of pN.²² A combination of eight lateral solenoids with an axial optical trap can produce much larger forces and torques and boasts very high bandwidth force measurement.²³ A design similar to the one described herein, with only two pairs of coils, produced bidirectional control and improved resolution of the force by placing one pair above and the other below the sample.²⁴

Overall, electromagnetic tweezers are founded on relatively basic concepts in electromagnetics and particle motion through a viscous fluid. Furthermore, they can be constructed inexpensively using a combination of commercial, off-the-shelf items and custom parts constructed with relative ease within university environments. Here, we describe a simplified electromagnetic tweezer design using two pairs of solenoids in an orthogonal arrangement (Fig. 2). We demonstrate that this implementation can manipulate MyOne paramagnetic beads with forces of over 15 pN. This instrument leverages manufacturing equipment that is increasingly available in university makerspaces and student machine shops. Thus, our eMT not only provides a system to manipulate a wide range of biopolymers and associated enzymes for vanguard research, but also a powerful teaching tool for biophysical education.

FIG. 2.

Field orientation. Coils on opposite corners of the square pattern are wired in series to align the fields between the pole pieces that extend below. Modulating the magnitude and polarity of currents through these pairs of solenoids generates fields (red or blue arrows) that combine vectorially to create a net magnetic field (green arrow).

III. DESIGN

As an instrument for undergraduate education in biophysics, we attempted to devise an eMT that met the following criteria. First, the construction of the apparatus utilized components and/or manufacturing tools generally available within physics and engineering departments. For instance, the design of the electromagnet iron cores was constrained by whether the geometries could be machined in their entirety using manual lathes and mills. Second, the electrical control hardware was based on popular open-source physical computing platforms with low barriers to entry, such as Arduino. In addition, we ensured that any custom electrical hardware for this project could be manufactured using a number of different techniques often available to undergraduate students. Furthermore, as budgets for undergraduate projects are often limited, we attempted to build an instrument that is relatively inexpensive. Overall, we estimate that the electromagnet hardware for this instrument—that is, excluding the optical microscopy components—can be constructed for under 200 US dollars.

An equally important constraint is the capability to generate forces on a similar scale to those produced by typical DNA-interacting enzymes. An adequate force range for an MT study should include magnitudes that stall molecular machinery or produce conformational changes. In a previous study, prokaryotic RNA polymerase was shown to exert forces $>10$ pN on DNA during transcription.²⁵ For studies that include DNA constructs with hairpins or cruciforms, forces $>15$ pN are often required to induce conformational change.¹⁸ DNA supercoiling behavior is also highly dependent on the tension in the molecule. The response of DNA to over- and under-wind differs between low ( $\mathrm{F}<0.5$ pN), intermediate ( $0.5\mathrm{pN}<\mathrm{F}<3$ pN), and higher ( $3\mathrm{pN}<\mathrm{F}<10$ pN) forces.²⁶ Therefore, the capability to quickly modulate force in the range of 0–20 pN is ideal.

The response of a paramagnetic bead (PMB) with mass $m_b$ and velocity ${\mathbf{u}}_b$ in a liquid exposed to an magnetic field gradient is given by

$$m_b\frac{d{\mathbf{u}}_b}{dt}={\mathbf{F}}_D+{\mathbf{F}}_M,$$ (1)

where ${\mathbf{F}}_D$ is Stoke’s drag and ${\mathbf{F}}_M$ is the magnetic force. The drag force ${\mathbf{F}}_D$ on a bead with radius $r$ suspended in a medium with dynamic viscosity $\eta$ is

$${\mathbf{F}}_D=-6\pi \eta r{\mathbf{u}}_b.$$ (2)

The magnetic force on the PMB depends on the negative gradient of the magnetic energy $U$,

$${\mathbf{F}}_M=-\mathrm{\nabla }U=\frac{1}{2}\mathrm{\nabla }(\mathbf{m}(\mathbf{B})\cdot \mathbf{B}).$$ (3)

In Eq. (3), $\mathbf{B}$ is the magnetic flux density and $\mathbf{m}(\mathbf{B})$ is the magnetic moment of the PMB, which depends on the external magnetic field. For small relatively weak fields, the force on the PMB scales linearly with gradient of the square of the magnetic flux density and the difference in magnetic susceptibility $\mathrm{\Delta }\chi ={\chi }_b-{\chi }_a$ between the bead material and ambient medium (one can often neglect the ambient susceptibility because it tends to be several orders of magnitude smaller than that of the PMB),

$${\mathbf{F}}_M=\frac{V_b{\chi }_b}{2{\mu }_0}\mathrm{\nabla }|\mathbf{B}{|}^2,$$ (4)

where $V_b$ is the bead’s volume and ${\mu }_0$ is the permeability of free space. For larger fields, conversely, the magnetic moment of the bead reaches a saturation value ${\mathbf{m}}_{\mathrm{sat}}$ and the expression for magnetic force reduces to

$${\mathbf{F}}_M=\frac{1}{2}\mathrm{\nabla }({\mathbf{m}}_{\mathrm{sat}}\cdot \mathbf{B}).$$ (5)

Micrometer-sized beads reach terminal velocity very rapidly in the viscous media and the acceleration term on the left-hand side of Eq. (1) goes to zero. Thus, the motion of a bead is governed by the balance between the magnetic force and viscous drag,

$$6\pi \eta r{\mathbf{u}}_b=\frac{1}{2}\mathrm{\nabla }({\mathbf{m}}_{\mathrm{sat}}\cdot \mathbf{B}).$$ (6)

Implicit in Eq. (5) is the fact that the force on the bead depends on the magnetic field gradient, not on the magnitude. Thus, an effective eMT requires an electromagnet capable of producing a highly non-uniform magnetic field at the position of the tethered bead. For instance, to obtain trapping forces on the order of tens of pico-Newtons on 1  $\mu$m Invitrogen MyOne requires magnetic flux density gradients on the scale of ${10}^2$– ${10}^3{\mathrm{Tm}}^{-1}$.

The torque $\tau$ is given by

$$\tau =\overrightarrow{m_o}\times \overrightarrow{B}.$$ (7)

Above, $m_o$ is a component of magnetic moment not aligned with the external field. Unlike the force, $\tau$ depends on the magnitude of the magnetic flux density.

After considering various eMT configurations, we converged on a design that includes four solenoid coils arranged vertically at the four corners of a square frame (shown schematically in Fig. 2). In this configuration, solenoids on opposite corners are wired in series, with one coil wound in a clockwise sense and the other wound counterclockwise. Each solenoid consists of $\sim 460$ turns of polyimide-insulated copper wire wound onto tapered, low carbon steel cores (obtained from McMaster-Carr, part 8910K77 or similar). By varying the current through the solenoid pairs, the horizontal component of the resulting $\mathbf{B}$ vector can be rotated ${360}^{°}$ in the xy plane. Furthermore, the pole tips are angled downward, meaning the field has a vertical component to attract a bead. As noted above, the force on a bead requires that this field be highly non-uniform. We achieve this by tapering the pole pieces down to sharp tips at the center of the quadrapole.

Prior to manufacturing the proposed eMT system, we estimated its performance to a first order using Elmer, an open-source finite element analysis (FEA) solver developed at CSC-IT Center for Science (CSC: https://www.csc.fi/web/elmer). Drafting and meshing of the model geometry were done using gmsh (https://gmsh.info/), which allowed for flexible mesh refinements, especially in regions of high field gradients. We simulated a single pole pair attached to a yoke, corresponding to the case in which a bead’s moment is aligned with the pair’s axis (vector ${\mathbf{B}}_{13}$ in Fig. 2). Simulations were conducted in full 3D for coil currents in the range of 0.3–3 A. The distribution of magnetic field lines for a coil current of 1 A is shown in Fig. 3 (see the supplementary material for more information on model setup).

FIG. 3.

Finite element modeling. Distribution of magnetic flux density as computed with Elmer FEA. This rendering represents simulations with the coil current set to 3 A. Brighter polygons in the middle of the renditions are artifacts of visualization software. The z-axis, pointing toward the sample, is represented by a white arrow.

Assuming a PMB is located along the $z$-axis (a white arrow in Fig. 3), we can estimate both the magnetic flux density and the gradient for a given solenoid current. These two quantities are shown in Figs. 4(a) and 4(b), respectively. Using Eq. (5), we can also estimate the range of forces our design can produce on a PMB. Here, we consider MyOne Streptavidin T1 (Invitrogen, Grand Island, NY) and MagSense (MagSense, IN, USA) beads. According to the manufacturer, 1  $\mu$m MyOne beads have a specific magnetization at a saturation of $20\mathrm{A}{\mathrm{m}}^2$/kg, meaning that ${\mathbf{m}}_{\mathbf{s}\mathbf{a}\mathbf{t}}\approx 4\times {10}^{-14}\mathrm{A}{\mathrm{m}}^2$. For 1.05  $\mu$m MagSense beads, this value is around $\approx 7\times {10}^{-14}\mathrm{A}{\mathrm{m}}^2$. Given these bead properties and the field gradients rendered in Fig. 4(b), we expect to be able to exert forces of up to several tens pN on both classes of beads. This is shown explicitly in Fig. 5, where we plot the force on MyOne [panel (a)] and MagSense (b) beads as a function of both driving current and distance from the pole tips of the eMT.

FIG. 4.

Magnetic field parameters. (a) Magnetic flux density as a function of solenoid current and distance from the pole tips along the z-axis (see the white arrow in Fig. 3). (b) Gradient of the magnetic flux density as a function of distance from the pole tips.

FIG. 5.

Estimated forces on beads. Forces on beads as a function of distance from the electromagnet pole tips and solenoid current. Forces were computed using Eq. (5) for two classes of beads: (a) 1  $\mu$m Invitrogen MyOne and (b) 1.05  $\mu$m MagSense. The red arrows in (a) indicate positioning of the physical electromagnet from the sample in experiments described below.

IV. IMPLEMENTATION

The high-level block diagram shown in Fig. 6 highlights the principal subsystems composing our eMT: the solenoid quadrapole, driving electrical hardware, and software.

FIG. 6.

System summary. Three main components were integrated for the electromagnetic tweezer. A GUI developed in MATLAB issued commands through a USB interface to custom electrical circuitry. This circuitry includes an Arduino microcontroller regulating two H-bridge devices that distribute current from a DC power supply to solenoids at opposite corners of the electromagnet frame.

A. Solenoids

As noted above, we manufactured each solenoid by winding 460 turns of polyimide-insulated copper wire onto tapered, iron cores (Fig. 7). Care was taken to ensure that the wire was wound with constant tension. A principal motivation for selecting the design was the machinability of the iron cores. Specifically, we were able to fabricate the cores using solely a manual mill and a set of angle blocks. Such tools are commonly available in student machine shops. The quadrapole was then mounted onto a bright-field microscope previously described in Kovari et al.²⁷ We machined the iron cores based on the geometries we used for the simulations described above. These geometries can be found in the supplementary material.

FIG. 7.

Electromagnetic tweezer prototype. (a) Computer aided design (CAD, Autodesk Fusion 360 student licence) rendering of electromagnetic tweezers (bottom view), showing octahedral pole-pieces made from soft iron. The tapered ends increase the field gradients and converge to superimpose at the center. (b) Oblique top view showing the four wire solenoids wound onto iron cores. Solenoids are mounted onto a soft iron yoke (dark gray), which reduces stray fields. The yoke screws on aluminum block with a cylindrical central bore (light gray), which provides a mechanical interface to the microscope (not shown). (c) Photograph of the prototype eMT in position above the specimen stage of the microscope. 625 nm light from a LED mounted above the eMT (not pictured) passes through the $\sim 1$ mm gap in the pole pieces to illuminate a sample taped to the microscope stage. Beneath the stage (not shown), a 63X objective broadcasts an image through a tube lens onto a camera (see Kovari et al. for details).²⁷

B. Electrical hardware

To power the solenoids, we developed the custom electrical hardware shown schematically in Fig. 8(a). This control circuitry was built around an Arduino-compatible microprocessor [either Arduino Metro Mini or Arduino Micro; see Figs. 8(c) and 8(d)]. Arduino was selected given its ubiquity in classroom and makerspace environments. The processor serves to control a set of H-bridge circuits, which set current direction and magnitude through the coils. Because H-bridges are employed widely to drive inductive loads (primarily DC motors), they are readily available as monolithic integrated circuits (ICs) capable of handling large currents. Our eMT uses ST VNH5019 series automotive H-bridges, which can deliver continuous currents of 12 A (with appropriate heat sinking) and 30 A peak current (note: early prototypes used VNH2SP30, but this component has since reached end-of-life). The VNH5019 is popular in the hacking and maker communities, because it easily interfaces with popular microcontrollers through 2.5–5 V logic levels. Each VNH5019 requires three digital connections from the Arduino processor: (1) An enable (EN) line which activates the H-bridge, (2) a direction (DIR) line which sets the voltage polarity across a solenoid (and, thus, sets current direction), and (3) pulse-width modulation (PWM) signal to set the effective power delivered [Fig. 8(a)]. Additionally, each H-bridge requires a 2.5–5 V supply to drive internal logic circuity and a 5.5–24 V rail capable of sourcing up to 3 A to power the inductive load (i.e., the solenoid).

FIG. 8.

Electrical hardware. (a) Schematic diagram of one of the solenoid circuits. Each solenoid is driven by a VNH5019 monolithic H-bridge. The device is controlled by three logic lines from a microcontroller (here, we used Arduino-based systems) and accepts a wide range of input voltages (5.5–24 V). With appropriate heatsinking, the H-bridge can source up to 12 A. (b) Photograph of a PCB manufactured using a table-top milling machine. (c) Photograph of a two-channel prototype using patterning and chemical etching. (d) Photograph of a completed four-channel prototype [the same design as in (b)] manufactured by a professional PCB fabrication house.

To provide functional flexibility, our design implements four H-bridge ICs, which can be configured in two ways. In one configuration, opposing pole pieces are connected in series to a single H-bridge IC. In this mode, the two unused H-bridge ICs may be employed to power active cooling devices to prevent overheating of the solenoid hardware and sample under test. In the other configuration, each VNH5019 drives a single solenoid coil independently. This allows compensation for any operational mismatch between solenoids. For the experiments discussed below, we used the first mode.

We made electrical connections using a printed circuit board (PCB) drafted in KiCAD (files available in the supplementary material), an open-source electronic design automation suite available for Linux, Mac, and Windows (https://www.kicad.org/). KiCAD supports multilayer-board designs, meaning that electrical traces can be routed on both the top and bottom sides of the board as well as internally through the board.

Given the variability of electronic fabrication resources on university campuses, we considered a circuit layout compatible with a wide range of manufacturing procedures. “In-house” production may involve milling or etching copper-clad FR4 sheets, processes which typically allow for two-layer designs (top and bottom) only. Milling, where copper traces are formed by mechanically removing portions of the superficial copper layers, is the simplest of the two. However, it requires a table-top milling machine or a dedicated PCB router. We produced an initial two-layer prototype of the driver board using an LPKF Protomat S104 [see Fig. 8(b)]. The second “in-house” process, etching, may require less specialized equipment, but involves more steps, practice, and potentially hazardous chemicals (i.e., presenting danger to the user and requiring special disposal). First, a temporary mask representative of the board design is patterned onto the copper laminate. The board is then placed in an acid bath, which removes the copper not protected by the mask, leaving the desired electrical pattern behind. Finally, the mask can be removed with solvents. An example of a board produced using etching is rendered in Fig. 8(c). Fritzing, an online open-source hardware collective offers an excellent tutorial on PCB etching (https://fritzing.org/learning/tutorials/pcb-production-tutorials/diy-pcb-etching).

Lastly, printed circuit boards may also be manufactured through a professional fabrication house offering student or academic discounts. An advantage of this strategy is the fact that fab houses can produce high quality boards with more than two copper layers. Thus, in addition to routing power and signals on the top and bottom layers for a final prototype (as done with the two other methods), we defined two internal layers for thermal management. These layers consisted of large copper areas (pours) that funnel heat from the H-bridges to the circuit board’s enclosure via aluminum standoffs. Figure 8(d) shows a photograph of a professionally fabricated board by Advanced Circuits (Aurora, CO).

Being rather small, components (H-bridges, resistors, capacitors, etc.) were soldered onto the boards using a reflow oven built from a modified conventional toaster oven (instructions on how to build one can be found here: https://learn.adafruit.com/ez-make-oven/the-toaster-oven). Overall, we found that the three manufacturing techniques were adequate for PCB production in the context of this eMT. For milled and etched boards, however, care must be taken to ensure heat is dissipated effectively away from the H-bridge ICs when driving the solenoids with high current. If PCB is manufactured with these “in-house” techniques, we suggest the use of large heat sinks and that two of the four H-bridge ICs be used to drive active cooling elements as described by Jiang et al.²¹

C. Software

Beyond driving the solenoids, the Arduino-based processor serves as an interface between the eMT and a desktop computer. The processor and the computer exchange information through simple serial strings across USB. The commands, which set the eMT’s field strength and xy orientation, may be sent through the serial monitor included in the Arduino Integrated Development Environment or any other serial terminal program (see the supplementary material for a list of commands). To make the system more user friendly, however, we integrated this functionality into a MATLAB graphical interface which also captures images of the tethered PMB from the microscope’s camera and leverages previously described algorithms to analyze bead (tether) dynamics, for instance, extension and force.²⁷

V. TESTING

A. Optical components

We tested the operation of the electromagnetic tweezer using a previously described²⁷ microscope developed for tethered bead tracking. A picture of this setup can be found in Fig. 9. The microscope is built around an Oriel Corporation optical post (Stratford, CT). A 635  nm LED light emitter (Mouser Electronics) fixed into a custom-made collimator begins the optical train. The light travels down through the microscope axis and illuminates the sample placed on the stage through the gaps in the pole pieces. A piezoelectric objective mount (Pifoc/Physics Instruments, Germany) allows positioning of a 63X objective (Leica, Germany). After traveling through the objective, light from the sample is focused by a relay lens onto an acA2000-165um-Basler ace camera (Balser, Exton, PA). Finally, the camera passes live video information to the computer and MATLAB tracking software.

FIG. 9.

Magnet tweezer tower. (a) A photograph of the custom electromagnetic tweezer with labeled components.

B. Particle tracking, tether length, and tension

Most magnetic tweezers are based on optical microscopy with real-time or offline analysis of diffraction patterns to track beads and determine tether lengths. Light and beads of a similar scale generate circular diffraction patterns of the beads (Fig. 10) that vary according to the distance of the bead from the focal plane. Our magnetic tweezers utilize image analysis to locate points of radial symmetry and determine x and y image coordinates for each bead. Time averages of these coordinates identify the anchor point. Then, the z position is determined by the best match between the radial intensity profile of the diffraction pattern and those recorded previously for the same beads over a range of focal offsets of the objective.^15,27,29 For a time interval with n recorded images, the average, effective tether length $⟨L⟩$ is a function of the recorded $\delta x$, $\delta y$, and $\delta z$ bead excursions about the anchor point in each video frame,

$$⟨L⟩=\frac{1}{n}\sum _{i=1}^n{(\delta x_i^2+\delta y_i^2+\delta z_i^2)}^{-1/2}.$$ (8)

FIG. 10.

Bead tethering and tracking. (a) A diagram of a DNA-tethered bead. The DNA is anchored to the glass at the origin of a Cartesian coordinate system and the coordinates $\delta x$, $\delta y$, and $\delta z$ represent the position of the bead. (b) A representative image from the microscope showing the circular diffraction patterns of DNA-tethered MyOne beads. The x and y coordinates are established by locating the convergence of local intensity gradient vectors,²⁸ while z is calculated from a correlation of diffraction patterns vs displacements of the objective.²⁷

Based on the equipartition theorem, the force on the bead during an interval can be calculated as a function of the time-averaged effective tether length $⟨L⟩$^1,15 and the variance of bead excursion along the magnetic field’s axis $⟨\delta y^2⟩$,

$$⟨F⟩=\frac{k_{B}T⟨L⟩}{⟨\delta y^2⟩},$$ (9)

where $k_B$ is Boltzmann’s constant and T is the temperature of the system.

C. Sample choice and preparation

Although the folding of proteins, such as talin that plays a prominent role in cell adhesion³⁰ or the hemostatic von Willibrand factor,³¹ are being investigated using magnetic tweezers, MTs are more frequently utilized to twist and stretch DNA to examine elasticity, topology, and processing by enzymes. This is in part because protein unfolding may require forces beyond the typical range (tens or hundreds of picoNewtons) available in magnetic tweezing and the ease with which DNA specimens can be prepared using molecular biology. Our lab investigates DNA, so we prepared microchambers with DNA-tethered beads as specimens with which to characterize the electromagnetic tweezer. Coverslips were cleaned in lab soap for 60 min with gentle oscillation, after which they were rinsed copiously with water and distilled water and immersed in ethanol for storage. To assemble a microchamber, a laser-cut, parafilm gasket with a central opening slightly longer than 22 mm was placed between cleaned $24\times 50$ and $22\times 22$ (mm) coverglasses, and the assembly was heated briefly for sealing. The approximately 15  $\mu$l chamber created by the cavity in the gasket between the glasses was rinsed with phosphate-buffered saline (PBS) and incubated with 4  $\mu$g/ml anti-digoxigenin (Roche, Indianapolis, IN) in PBS for several hours at room temperature or overnight at $4^{°}\mathrm{C}$. After incubation, the chamber was rinsed with West-Ez Blocking Buffer with 1% Casein (GenDEPOT, Barker, TX) and incubated for 30 min.

A short DNA tether consisted of a 2036 bp central fragment produced by PCR with forward primer $5^{\prime }$-acccgtgggcccAGCATCCTCTCTCGTTTCATC and reverse primer $5^{\prime }$-gcagcgcccgggTGAGCGAGGAAGCGGAAGAG with a pZV_21_400³² plasmid template. DNA segments for anchoring the central fragment were similarly amplified with forward primer $5^{\prime }$-GGCGATTAAGTTGGGTAACG and reverse primer $5^{\prime }$-TGTGGAATTGTGAGCGGATA using the pBlueScript plasmid template to create a 302 bp segment centered on the multicloning site, but 10% of the dTTP nucleotide in the PCR was replaced by biotin-16- (Thermofisher, Waltham, MA) or digoxigenin-11-labeled dUTP (Roche, Indianapolis, IN) to yield biotin- or digoxigenin-labeled fragments. The central fragment was digested with ApaI and XmaI and the labeled segments with either ApaI or XmaI to produce approximately 100–200 bp biotin- and digoxigenin-labeled tail fragments with complementary ends for ligation. The products were then purified and ligated using the T7 DNA ligase. Enzymes for molecular biology were purchased from New England Biolabs (Ipswitch, MA).

A slightly longer DNA tether with a 2098 bp central fragment was also produced by PCR using forward primer $5^{\prime }$-GGCGATTAAGTTGGGTAACG and reverse primer $5^{\prime }$-TCAAATAAATTTCCGCTCATGAGACAATAA with the pUC19 plasmid template. DNA segments for anchoring the central fragment were similarly amplified with forward primer $5^{\prime }$-ATAGTTACCGGATAAGGCGCAGCG and reverse primer $5^{\prime }$-TGGGTGAGCAAAAACAGGAAGGCA to create a 1476 bp segment centered on the multicloning site, but 10% of the dTTP nucleotide in the PCR was replaced by biotin-16- (Thermofisher, Waltham, MA) or digoxigenin-11-labeled dUTP (Roche, Indianapolis, IN) to yield biotin- or digoxigenin-labeled DNA. Subsequently, the central fragment was digested with ApoI and HindIII. The labeled segments were digested with either ApoI or HindIII to produce approximately 740 bp biotin- and digoxigenin-labeled tail fragments with complementary ends for ligation. The products were then purified and ligated using the T7 DNA ligase.

A third DNA tether with a 2580 bp central fragment was also produced by PCR using forward primer $5^{\prime }$-GTAAAACGACGGCCAG and reverse primer $5^{\prime }$-TGGCGTAATAGCGAAGAG with the pUC19 plasmid template. Biotin- and digoxigenin-labeled fragments were produced as above. Subsequently, the central fragment was digested with AvaI and KasI enzymes, while the 1476 bp labeled fragments were digested with either KasI or AvaI, respectively. The products were then purified and ligated using the T7 DNA ligase.

To assemble the DNA-tethered beads, a 1  $\mu$l aliquot of streptavidin-coated, 1  $\mu$m-diameter magnetic beads (MyOne Streptavidin T1, Invitrogen, Grand Island, NY) was washed 2X with 1 molar NaCl and mixed with the DNA construct in 100 mM NaCl, 20 mM Tris-HCl pH 7.4, 1 mM EDTA. Following a 5-min incubation, the DNA–bead mixture was added to a pre-assembled flow-chamber and incubated for 10 min. Unbound DNA and beads were flushed out with 100 mM NaCl, 20 mM Tris-HCl pH 7.4, 1 mM EDTA, 0.5% Tween, 0.2 mg/ml $\alpha$-casein (2580 bp tethers), PBS with 1 mM EDTA and 0.1% Tween (2098 bp tethers), or 10 mM Tris-HCl pH 7.4, 200 mM KCl, 0.1 mM EDTA, 5% DMSO, 0.2 mM DTT, 0.2 mg/ml $\alpha$-casein (2036 bp tethers) before brief storage at $4^{°}$C or observation. The buffer was chosen for convenience and did not affect the data significantly.

D. eMT settings

We conducted all experiments using the serial configuration for the solenoids—that is, a single H-bridge driver powered a pair of solenoids connected in series. We set the potential across the H-bridges to 12 V. Given that the series resistance of the solenoid pair is around 4  $\mathrm{\Omega }$, the maximum current in the solenoid pair was on the order of 3 A. As mentioned above, however, the H-bridge can deliver smaller amounts of power to the solenoids depending on the duty cycle of the PWM signal from the microcontroller (e.g., 0% duty cycle corresponds to 0 A, whereas 100% duty cycle provides the full 3 A). The Arduino board we used in these tests is based on an Atmega328 which has 8-bit PWM hardware. Thus, we could change the current through the solenoids with a resolution of 0.011 A.

After mounting the eMT on the microscope displayed in Fig. 9 and placing the microchamber onto the stage, we lowered the eMT so that the pole pieces were $\sim 1$ mm above from the top coverslip of the chamber. Using images from the camera, a field of view was selected that included both tethered beads exhibiting constrained Brownian motion and beads stuck to the glass. After selecting small regions of interest (ROI) for individual particle tracking, time series of ROIs were recorded under high force, aligned, and averaged for each bead at a series of focal offsets of the objective, which was mounted on a piezoelectric positioner (PIFOC, Physik Instrumente, Auburn, MA). Subsequently, during the experiments with the objective position held constant, this calibration scan served as a look up table to determine z based on changes in the diffraction pattern.²⁷

VI. RESULTS AND DISCUSSION

A. Stretching the DNA tether

A force vs extension experiment was used to determine the range of forces generated with the electromagnet at different distances from the 2098 bp tethers. The field was oriented at $0^{°}$, corresponding to equal contributions of current/field through the orthogonal solenoid pairs. The force setting was increased from 0 and 100% of maximum current (3 A) in steps of 10%, 10 s per interval.

There is a clear negative correlation between generated force range and eMT height above the sample [Fig. 11(a)]. The maximum attractive forces with 3 A of current at the various eMT positions correspond well to those predicted by the model (purple). As expected for a worm-like chain, Fig. 11(b) shows that the compact, randomly coiled DNA extended by large increments in response to current changes at low levels. Beyond that, with both DNA tethers more extended, similar steps of the current produced smaller length increases. The force was calculated using the equipartition theorem expression [Eq. (9)] described by Vilfan et al.¹⁵

FIG. 11.

DNA tether response to the applied electromagnetic force. (a) The attractive force exerted on the DNA is highly dependent on the distance of the electromagnet from the PMB. The curves represent different 2098 bp DNA tethers that extended as the current was increased from 0 to 3 A at 10% intervals. At shorter separations, such as 0.8 mm (red) and 1.3 mm (yellow), the maximum current generates forces up to 10 pN. When the eMT distance was increased to 2.8 mm (blue), the force range decreased to a sub-pN magnitudes at maximum current. The error bars are calculated as a function of standard deviation of tether length and y variance. For reference, the estimations of force predicted by the numerical model for a current of 3 A at distances of 0.8, 1.3, 1.8, and 2.8 mm [red arrows in Fig. 5(b) are indicated by purple symbols]. These estimations show reasonably good agreement with experimental data. (b) Under increasing tension, the randomly coiled DNA tether extends, leading to a greater effective tether length. In accordance with the worm-like chain model, tensile forces $>2$ pN induce maximum extension of the randomly coiled DNA polymer. Several force ranges were explored via different eMT heights, and the maximum tether length was consistent with the expected length of a 2098 bp tether (0.7133  $\mu$m $+/-$ 10%).

Additional trials with longer DNA constructs demonstrated maximum forces slightly above 15 pN and no remarkable hysteresis. See the supplementary material for data.

B. Twisting the DNA tether

Under low tension, rotating a tethered bead twists the attached DNA and increases the torque in the DNA tether. Twisting beyond the critical buckling torque causes the molecule to curl, form a loop and finally a plectoneme, which reduces the overall extension of the DNA. For this experiment, the pole pieces were positioned 1.8 mm above DNA tether, and at constant current (force), the electromagnetic field was turned to rotate the bead and twist the attached DNA. While tracking beads at a rate of 50 Hz, the field was rotated by +0.05 turns every 0.1 s. Video tracking was undisturbed by rotation of the electromagnetic field and raw data points clustered along the moving average as shown in Fig. 12(a). Plectonemes absorbed either positive or negative turns at low force and reduced the tether length. Figure 12(b) shows moving averages of tether lengths produced by twisting at a series of different forces. At higher forces, negative turns were partially absorbed as twist and the tether length did not contract to the same degree. This asymmetric response to twist is well documented in the literature.^26,33 With no mechanical vibration associated with changing the orientation of the B field, tether lengths change during twisting could be monitored without interruption.

FIG. 12.

DNA tether length vs applied twist as a function of force. The extension of a 2036 bp DNA tether, z position of the bead was monitored continuously as the DNA was twisted. (a) Undisturbed video tracking produced uninterrupted data (purple data points and a blue moving average) through a full range of twist that contracted the tether into plectonemes for both positive and negative turns at low tension. (b) At higher forces, negative turns were partially absorbed as twist without contraction as plectonemes. The measured forces for each twist curve are indicated in the legend.

C. Rapid force loading

Tension along the double helix modulates the twist-writhe partitioning of a DNA molecule and influences energetically favorable conformations. For investigating conformational changes, an instrument capable of high frequency force modulation with high temporal resolution is desirable. With an electromagnet, the magnetic field can be abruptly changed by altering the current with no mechanical motion required. Therefore, a stepwise test (Fig. 13) was conducted stepping between current values of 0% to 20%, 40%, 60%, 80%, and 100%, for 15 s intervals. Pole pieces were positioned 1.8 mm above the sample. Comparison with the magnitudes of forces produced by more incremental changes as shown in Fig. 11 reveals no remarkable hysteresis on this time scale. Indeed the change from 0 to 100% generated with either paradigm of increasing force produced asymptotic tether length and force plateaus. Transitions between low and high force states occurred in less than 1 s with no spurious tether lengths that would indicate particle tracking errors.

FIG. 13.

Rapid current modulation produced negligible hysteresis. (a) Increasingly larger steps in the current extended a 2580 bp molecule asymptotically. However, the molecule quickly retracted to the initial average length when the current suddenly dropped to the initial value. (b) Increasingly larger steps in the current produced increasingly larger changes in force with a constant baseline indicating negligible hysteresis. Even large changes in force can be monitored without interruption with electromagnetic modulation.

Note, however, the non-zero forces at 0% current, which indicate remnant fields in the pole pieces. These residual forces emphasize that materials, like the low carbon steel (iron) in the solenoid cores, are not perfectly paramagnetic. Once aligned by an externally applied field, magnetic domains in the material may not randomize completely when the external field is extinguished, leaving a remnant field. Such hysteresis might limit the minimum force in studies where forces are modulated in a stepwise manner. Although some degree of hysteresis exists, the pole-piece distance may be changed to mitigate the effects of remnant fields. This test indicates negligible delays associated with abruptly modulating the electromagnetic field strength.

Upon abruptly decreasing the current, the force dropped repeatedly to near baseline values equal to the initial value with small standard deviations regardless of the magnitude of the current drop. However, the standard deviation increased slightly for larger changes, indicating slight hysteresis that may affect large, rapid force increases.

Slower modulation of the current also produced low hysteresis. Figure 14 shows three representative, 2098 bp DNA tethers that were monitored as the current was raised in 10% steps at 10 s intervals from 0 to the maximum (3 A) and lowered again. As shown, the forces on the beads at identical current settings generated similar tensions during these slowly rising or falling current sequences.

FIG. 14.

Slow current modulation produced low hysteresis. For three different molecules, the current was increased by 10% every 10 s up to the maximum and then decreased in 10% steps every 10 s until reaching zero. The resultant tensions produced on the molecules displayed little hysteresis.

VII. CONCLUSIONS

An electromagnet that exerts as much as 15 pN of force on 1-  $\mu$m diameter, super-paramagnetic beads was realized on the basis of a conceptually simple design. This is significant because the MyOne beads are small in comparison to those used in several other studies with electromagnets. Smaller beads experience less drag and exhibit rapid Brownian diffusion, which improves temporal resolution, but their size limits the maximum achievable forces. We have demonstrated that our design produces an ample range of physiologically relevant forces on 1-  $\mu$m-diameter, paramagnetic beads, and tenfold changes of the applied force can be achieved in ten ms or less without disturbing particle tracking. These features will facilitate single molecule investigations of polymers in biophysics. Furthermore, we constructed the system using low-cost hardware and software that is becoming increasingly available to students in university and college settings. As such, we hope this tool will find use in undergraduate biophysical education.

A number of improvements are planned for the electromagnetic tweezers in the future. We hope to develop an active cooling system to take advantage of the two unused H-bridge channels in our design. Such implementation will mitigate the effects of heat produced by the solenoids at high currents. Additionally, we hope to refine our numerical model to make it more accurate and user friendly. Other possible improvements include laminating the solenoid cores and switching to more paramagnetic eMT frame materials such as Mu metal. Ultimately, we hope to consolidate the electromagnetic tweezer into a smaller, portable system that can be integrated into a greater variety of microscopy systems in a plug-and-play fashion.

SUPPLEMENTARY MATERIAL

See the supplementary material for tools used to build the electromagnetic tweezers presented in the main text.

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

J.G.P. and J.M.H. contributed equally to this work and are co-first authors.

DATA AVAILABILITY

The data that support the findings of this study are openly available in Emory DataVerse at https://doi.org/10.15139/S3/DIUTD9, Ref. 34.