Force spectroscopy has become a powerful approach for studies of single molecules [1–4], macromolecular assemblies [5–7] and even whole cells [8–11]. This approach was pioneered by the invention of optical tweezers, which manipulate dielectric microscopic objects with the help of a focused light beam [12, 13]. Soon thereafter, magnetic tweezers were invented to allow the manipulation of paramagnetic beads using a gradient of magnetic field [14]. The two techniques have been developing head to head to achieve nowadays a remarkable level of sophistication. They have their own unique advantages but, largely, offer similar capabilities and require the same workflow.

The main advantage of optical tweezers is the ease with which single beads can be manipulated and carried around. In contrast, magnetic tweezers are at their best when working with a homogeneous force field. This set-up does not generate a three-dimensional (3D) force trap but allows for easy multiplexing where multiple beads can be observed and manipulated at the same time. In addition, the method does not require intense sample irradiation, accommodates multiple DNAs (or other macromolecules) and allows remarkable ease in the stretching and twisting of macromolecules. This offers unparalleled simplicity and versatility in the design of downstream applications. This review discusses the architecture and use of magnetic tweezers, highlights key requirements for experimental design and provides several examples illustrating their use.

The first magnetic tweezers were assembled in 1996 by Strick, Bensimon and Croquette who used them to explore elasticity of supercoiled DNA [14]. In the original design, the experiment is conducted within a glass flow cell, which is placed atop an inverted microscope with a pair of permanent magnets above it (Figure 1). The flow cell contains DNA molecules stretched between its surface and a paramagnetic bead, and the force is varied by moving the magnets with the help of a motorized drive. Later implementations developed more sophisticated flow chambers and magnetic pole assemblies.

In a typical experiment, the position of the bead is monitored in 3D. This is achieved by computer-assisted analysis of the bead image, which changes not only during horizontal but also vertical displacement of the bead (see below). The changes in DNA extension are recorded in real time, in response to the activity of an enzyme or changes in the applied force. The flow cell is connected to the rest of a microfluidic system, which can be as simple as a single reservoir, or as complicated as the user requires. A light source is usually placed atop of the microscope but can be relocated when needed. The image of the bead is captured using a charge-coupled device (CCD) camera. In recent years, complementary metal oxide semiconductor (CMOS) cameras have been gaining in popularity due to their generally faster response times and the resulting ability to track motion at higher frequencies. To complete the system, a computer program that records the real time motion of the bead and provides control over the magnets is required.

Magnetic tweezers can be assembled in a variety of configurations to match diverse experimental needs. The first question that needs to be answered at the design stage is about organization of the magnetic field. This is a key factor that defines performance of the tweezers. The generated force is proportional to the gradient of magnetic field and directed toward the area with the strongest field [1, 15–17]. As a result, magnetic tweezers display different properties depending on the configuration of the field. A single magnet cannot apply torque. When a pair of magnets is employed, the force is directed toward the gap between the magnets, and away from the sample (Figure 2A). For applications that require true capture of the bead and its translocation, a more complex arrangement is needed, with magnets placed around the bead, so that the strongest field can be created in its vicinity [18, 19]. Furthermore, the field strength in this case must be dynamically controlled to ensure efficient confinement of the bead, which necessitates the use of electromagnets.

Orientation of the field is another possible variable. Horizontal magnets induce horizontal magnetic dipole in the bead (Figure 2A). Such beads follow the magnets as the magnets are rotated. This configuration is required if one is interested in twisting macromolecules. Vertical orientation of the field (Figure 2B) produces a torque-free tether, which might be useful in applications where stretching and twisting need to be uncoupled [20].

Using NdFeB magnets and the widely used 2.8 μm beads, one can easily produce forces up to about 20 pN at distances of about 1 mm [10, 21]. This force is sufficient for many single molecule applications and often must not be exceeded since higher forces can induce unwanted conformational transitions in biological specimens [22, 23]. Higher forces can be produced by using bigger beads, reducing the separation between the magnets and the bead (i.e., using flow cells) and optimizing the geometry of magnetic poles. Such modifications, however, often come at a price. Bigger beads, for example, generate greater drag and reduce, therefore, the temporal resolution of the experiment [24]. Likewise, the use of microfabricated devices [18] can produce forces in nN range due to a dramatic decrease in the magnet-to-bead separation but increases the cost of each experiment. Moreover, this arrangement foregoes a significant advantage of conventional tweezers, the ability to conduct experiments at constant force, since variations in magnetic field within the field of view can no longer be neglected.

Of particular interest are electromagnetic tweezers, which employ electromagnets to generate the field [19, 25]. Such instruments use electric current to modulate the strength and direction of the magnetic field and can accommodate efficient feedback loops to generate stable force clamps. With fewer moving parts, reduced vibration and faster control over the magnetic field, electromagnetic tweezers expand possible applications of the technique. Their main disadvantages are a rather significant cost of integration of electromagnets into the tweezers, and the need to control hysteretic effects in the ferromagnetic core of the electromagnet.

Magnetic torque tweezers (MTT) [26] and electromagnetic torque tweezers (eMTT) [27] are newer additions to the field. They expand the capabilities of conventional instruments by allowing direct measurements of the applied torque. This is achieved by introducing a weak horizontal gradient of magnetic field so that the twisted molecules could produce detectable counteracting torque. Following proper calibration, the torque on such molecules could be derived from the misalignment of the rotated bead and the magnet (Figure 2C).

The heart of magnetic tweezers is a computer program that tracks the bead in 3D space and reports its position in real time. In a typical experiment, the bead is observed through a static oil immersion objective and its image projected onto a digital camera (Figure 1). To compensate for instrumental drift, one might choose to track simultaneously two beads, one of which is tethered to DNA and the other is stuck to the surface and serves as a reference. In this set up, detecting the horizontal location of the beads is quite straightforward. Most tweezers, however, apply force vertically, making vertical displacements the ones to track. Tracking is achieved by analyzing the image of the bead, which changes depending on bead proximity to the focal plane of the objective (Figure 2D).

Since the size of the beads is comparable to the wavelength of incident light, their images are significantly affected by diffraction [28, 29]. The use of coherent light for illumination accentuates the diffraction rings and thereby increases the effective size of the image. This in turn helps overcome the pixelation limit and increases the precision in locating the bead. These issues, however, rarely limit the experiment and become important only when using small beads or studying stiff tethers. As discussed below, such systems maximize suppression of thermal noise. As a result, the precision of locating the bead begins to affect the overall precision of the instrument.

The instant position of the bead is measured by comparison to a reference set of bead images [30]. These are collected for each bead prior to the experiment. During this initial calibration, the bead is typically held in place at a high force to reduce fluctuations, and the objective is translocated in the Z-direction. The generated stack of images is then processed to produce a reference curve that relates the appearance of the bead to its separation from the focal plane. Thereafter, the objective is held in place and the bead is allowed to fluctuate in response to applied forces. The displacement of the bead linearly relates to the displacement of the objective needed to produce the matching image, ΔZ_bead = ΔZ_obj·n₂/n₁, where n₁ and n₂ are the refractive indices of the immersion oil and the reaction buffer, respectively (Figure 2E). Using interpolative techniques, one can achieve a precision of the order of 1 nm.

Attachment of DNA to both the magnetic bead and the surface of the flow cell is essential for single molecule studies. This is achieved by combining together an unmodified DNA with derivatized DNA handles. Such handles can be prepared in a variety of ways. Perhaps the easiest way involves PCR using modified dNTPs. For the generation of a DNA tether, modified dUTPs such as amino-allyl dUTP, digoxigenin-dUTP (DIG-dUTP), or biotin-dUTP, can be incorporated by enzymatic polymerase chain reaction (PCR) [31–34], primer extension [35] or terminal transferase DNA extension [36]. While ligation of labeled PCR fragments remains a popular protocol, the choice of labeling strategy depends on the nature of experimental design and the required lifetime of the linkage.

The most common method for surface attachment of DNA utilizes the non-covalent interaction between DIG labeled DNA and an anti-digoxigenin antibody coated surface, and between biotin labeled DNA and streptavidin coated magnetic beads. At forces of about 10 pN, the average lifetimes of these interactions are 4 and 18,000 s, respectively [37]. The lifespan of these interactions decreases exponentially with increasing force [37] which significantly reduces the duration of the DIG/anti-DIG attachment and presents a major problem for experiments requiring large forces, such as those involving hairpins.

An alternative approach, in which DNA is covalently attached to both the surface and the magnetic bead, has been developed to achieve longer lifetime of binding. To covalently link DNA to the glass surface, a small fragment of DNA is labeled with amino-allyl-dUTP, digested and ligated to one end of the tethering DNA. Once labeled, the DNA is injected into a glass chamber coated with either silane-PEG-NHS (silane-polyethylene glycol-N-hydroxysuccinimide) [37], or coated with ethanolamine/PEG or NHS-PEG-COOH [34]. In all cases, the primary amine of the amino-allyl-dUTP will react with the coated surface to form a covalent bond. By simply exchanging DIG/anti-DIG binding for covalent attachment to the glass surface, the lifetime is increased dramatically from a few minutes to a few hours [34].

The force exerted by the tweezers cannot be determined from first principles if only because magnetization varies from one bead to another. Instead, the instrument relies on quantification of Brownian motion of the bead. To this end, the thermal motion of the bead is recorded at several positions of the magnets, quantified to derive the forces acting on the bead, and the measured forces are then used to construct an empirical interpolation function. Different interpolation functions have been employed for the task including high power polynomials [38] and the exponential function F = F_max exp(AZ_mag + BZ_mag²) [30]. In this analysis, the bead must be suspended in solution since its interactions with the surface restrict its motion and leads to abnormal readings [39].

In principle, the force can be calculated from the magnitude of the horizontal bead fluctuations, <δy²>. According to the Equipartition Theorem, α_y<δy²> = kT, where α_y is the horizontal stiffness of the trap, T is the temperature and k is the Boltzmann constant (Figure 3). For a bead tethered to the surface via a single linkage, the stiffness is proportional to the applied force [14], α_y = F/l, where l is the length of the tether, leading to the following relationship:

Thus, the force can be readily computed once the length of the tether and the variance of fluctuations are known. In practice, however, the utility of Equation (1) is limited due to external vibrations in the instrument, which must be filtered out from the recorded Brownian motion. The most robust way to achieve it is by fitting the fluctuations in the frequency domain.

A detailed description of such analysis and associated problems is presented in Berg-Sorensen and Flyvbjerg [40]. The approach is based on solving the underlying Langevin equation in the frequency domain. Assuming a harmonic force field, the power spectrum of the bead motion is found to be the Lorentzian:

where S(f) is the power spectrum of the bead motion, ζ is the friction coefficient for the bead, and f₀ = α_y/2πζ is the cutoff frequency of the oscillator. For a spherical particle with the radius r in a liquid with the viscosity η, ζ = 6πηr. Regardless to the shape of the bead, the friction coefficient is related to its diffusion coefficient, D, via the Einstein equation, D = kT/ζ. Integration of Equation (2) over the entire range of frequencies (from zero to infinity) yields Equation (1).

Analysis of Equation (2) leads to important experimental considerations. The shape of a Lorentzian is completely defined by two parameters, its cutoff frequency f₀ and the peak value, A = 4kTζ/α_y², which describes the magnitude of low frequency fluctuations, when f ≪ f₀. Both of them depend on the stiffness of the trap, α_y, and the diffusion coefficient of the bead. Only by multiplying the two can the contribution from viscosity be eliminated, A·f₀ = 2kT/πα_y. Therefore, both A and f₀ must be determined in order to derive the force, which requires fitting the power spectrum to the Lorentzian over a broad range of frequencies.

To achieve this, the recorded noise must include high frequency motion, when f ≥ f₀. Otherwise, the measured magnitude of Brownian motion would be underrepresented by an unknown factor. This, however, seldom poses a problem because magnetic tweezers produce fairly soft traps (characterized by low f₀). For a 3 μm DNA attached to a 3 μm bead and stretched with a 10 pN force, the cutoff frequency is on the order of 10 Hz. For such systems, fluctuations can be accurately recorded using a low end CCD camera. Certain applications, however, require suppression of the noise. This becomes especially important in studies of fast processes or those that involve small changes in DNA length. Such suppression is usually achieved by using higher forces, shorter DNAs and smaller beads [41, 42]. All these factors increase the frequency of fluctuations (Equation 2) and necessitate the use of CMOS or high speed CCD cameras.

DNA stretching by force is largely defined by its polymeric properties. In the simplest, freely-jointed model, a polymer is viewed as a chain of rigid segments connected by completely flexible links (Figure 4A). This model was first proposed by Werner Kuhn in his analysis of rubber elasticity [43] and still remains a cornerstone of polymer physics. The model contains only one parameter, the length of the segment, b, also known as the Kuhn length or statistical segment length. This length serves as an operational measure of polymer rigidity since stiffer polymers remain straight over longer distances and vice versa. The number of the segments can be immediately derived from the length L of the molecule as N = L/b. For long polymers, with N ≫ 1, the model predicts the well-known random coil conformation [44, 45].

Statistics of the segment within a random coil is the same as found in any stochastic process, such as diffusion or random flight. Thus, a random coil is characterized by massive fluctuations on the order of the size of the coil itself. The average size of the coil scales with the square root of the polymer length, <R²> = Lb, and is much smaller that the length of the molecule. Double stranded DNA, being naturally straight and fairly rigid, conforms well to the predictions of the model. The length of its Kuhn segment is about 300 bp [46]. Thus, most test molecules will contain at least several Kuhn segments.

The force applied to a polymer distorts the otherwise isotropic orientation of the segments. Extended molecules allow fewer microscopic conformations and generate, as a result, a purely entropic elastic force that resists stretching. The magnitude of the extension can be analytically calculated in the freely jointed model [44]. The distribution function of the segments in the constant force field follows the Boltzmann distribution:

where ΔG is the free energy of the segment in the force field, ψ is the angle between the segment and the direction of the force, b·cosψ is the projection of the segment onto the force, Fb cos ψ is the energy of the segment in the field, and 2π sinψ is the statistical weight of the conformations with the given angle ψ (Figure 4C). The average projection of the segments onto the force field, L_z, can be calculated as:

where L(x) is the Langevin function, L(x) = coth(x) − 1/x. Since all segments are equally influenced by the force, the relative extension, z = L_z/L, of the chain would be given by:

The result predicts that the relative extension of all polymers follows the same universal function of the product of the force and the polymer rigidity expressed as its statistical length. When comparing different polymers, the more flexible one requires proportionally higher force to achieve the same relative extension. One might also notice that double stranded DNA is a very soft spring. A force as low as 0.1 pN stretches DNA to 60% of its contour length. In the limit of low extensions, when Fb ≪ kT, Equation (5) reduces to a linear dependence

Equation (6) is analogous to Hooke's law, which describes the behavior of an ideal spring. Notably, the Langevin function can be approximated by a straight line up to fairly large extensions, when Fb ≈ kT. Thus, polymers retain their ideal spring behavior for extensions that dramatically exceed their unperturbed size.

Equation (5) poorly describes DNA elasticity at high extensions [47]. This occurs because the freely jointed chain model completely ignores local polymer curvature on scales shorter than the Kuhn length. Such fluctuations, however, significantly distort the shape of both short and highly extended polymers [48, 49].

A much better description of such conformations can be obtained using the worm-like chain model (Figure 4B). The model accounts for local curvature by postulating that the adjacent monomers of the chain bend relative to each other by an average angle θ = <θ²>^1/2, which does not change along the chain. This approximation is realistic since one should expect a constant energy associated with each degree of freedom, including bending. Thus, for harmonic fluctuations, α_b <θ²> = kT, where α_b is the bending rigidity. This equation applies to naturally straight polymers, for which <θ> = 0, and therefore, fluctuations of the bending angle are equal to the angle itself, δθ = θ − <θ> = θ. This model predicts that the correlation in the orientation of non-adjacent monomers in the chain exponentially decays with increasing separation between the monomers:

where u₁, u_n are unit vectors associated with the two monomers, u₁·u_n is their dot product, l is the distance between the monomers along the chain, l₀ is the distance between two adjacent monomers, and P is the persistence length. The expression for the end-to-end distance, R, applies both to short and long polymers:

For long polymers, with L ≫ P, <R²> = 2PL. This relationship resembles the aforementioned conclusions of the freely jointed model (<R²> = bL). Thus, the model predicts the same statistics as the freely jointed chain, provided that L ≫ P and b = 2P. For short DNAs, Equation 8 provides a much better estimate of DNA extension.

The persistence length is the primary parameter that defines extension of polymers by applied forces. The persistence length of double stranded DNA measured using single molecule stretching agrees well with numerous previous studies [48, 50, 51]. Its value is 45 nm at moderate salt conditions (i.e., above 50 mM NaCl or 1 mM MgCl₂), but significantly increases in diluted salt conditions. Double stranded RNA is about 20% stiffer; in 10 mM sodium phosphate solution, its persistence length was estimated as 64 nm, compared to 54 nm for double stranded DNA [52].

The problem of the worm-like chain under tension can be solved only numerically. In practice, data on DNA extension should be fit to the following empirical equation, which agrees with the solution to within 0.1%:

where a₂ = −0.5164228, a₃ = −2.737418, a₄ = 16.07497, a₅ = −38.87607, a₆ = 39.49944, a₇ = −14.17718 [50]. Notably, this equation neglects any volume interaction in DNA. Indeed, such interactions significantly affect conformations of random coils but are virtually non-existent in highly extended molecules.

High forces do not only overpopulate extended conformations of the polymer but also stretch the molecule itself. The elastic modulus (or rather its 1D equivalent) for DNA, K, is fairly high, given the DNA stability, about 1,000 pN [51, 53]. As a result, DNA extensibility becomes relevant only at high tension, when the molecule is about to undergo a force-induced conformational transition. The precision of existing data is insufficient to discriminate between conceivable structural models of extension. Instead, data can be approximated assuming a constant DNA extensibility and replacing z in Equation 9b with z − F/K [51].

Notably, polymeric DNA models discussed in the previous section prove insufficient when applied to stretching experiments using moderately short DNAs [54]. This occurs because DNA attachment to macroscopic objects imposes rather restrictive boundary conditions on the orientation of its ends. Moreover, rotational fluctuations of the tethered bead further restrict DNA conformations in favor of the more extended ones. Due to these factors, the apparent persistence length measured in stretching experiments displays DNA length dependence and is significantly underappreciated, even for molecules as long as several persistence lengths. In such cases, DNA extension must be modeled explicitly or analyzed in the framework of a computer intensive finite worm like chain model [54].

A signature feature of magnetic tweezers is their ability to quickly and precisely twist macromolecules. This allows researchers to alter DNA supercoiling during the course of the experiment. This in turn better mimics the cellular environment, where DNA is typically topologically constrained [55–58]. However, DNA behavior under tension somewhat differs from that in bulk measurements, when it is unconstrained, an aspect that must be taken into account during experimental design. We provide below a brief synopsis of DNA topology while more detailed reading is available in numerous excellent reviews [55, 59, 60].

The theory of DNA supercoiling was developed for circular DNA but can be readily extended to linear, yet torsionally constrained molecules (i.e., those whose ends do not freely rotate). To achieve this, the ends of the DNA must be attached to both the bead and the flow cell via multiple linkages, so that by rotating the bead, the molecule can be twisted. In such molecules, one strand winds around the other a certain number of times, which can be changed only by rotating the bead or transiently breaking the molecule and passing DNA segments through the resulting gate. This number of intertwining is called the linking number, Lk. In torsionally unstrained DNA, the linking number is directly related to the most stable secondary structure of the molecule. Such DNA is called relaxed, and its linking number is denoted as Lk₀. Lk₀ is inversely related to the DNA helical repeat, γ, which is the number of base pairs needed to make a full turn within the DNA double helix under given conditions. For DNA composed of N base pairs, Lk₀ = N/γ. The difference between Lk and Lk₀ describes the extent of torsional deformation of the molecule and is called supercoiling, ΔLk = Lk − Lk₀, whereas superhelical density, σ = ΔLk/Lk₀, specifies a length independent measure of supercoiling. Inside bacteria, DNA is partially underwound, with σ ≈ −0.05 [61, 62]. Higher levels of torsional stress induce formation of non-canonical structures, such as stretches of melted DNA or Z-DNA (reviewed in [60, 63, 64]).

The linking number can be realized in the form of twist, Tw, and writhe, Wr. These two properties describe DNA topology by approximating it as a ribbon [65, 66] (Figure 5). By convention, the plane of the ribbon is associated with DNA grooves and the edges with the sugar-phosphate backbone. Twist is defined as the rotation of the edges of the ribbon around its axis and writhe describes the anisotropy of the axis in 3D space. Twist is a local property that can be readily related to the local structure of a molecule. It equals the sum of all local rotations between adjacent base pairs measured in turns. In relaxed DNA, Tw₀ = N/γ. By contrast, writhe describes the global layout of the chain. It is a measure of chirality and equals zero for flat molecules or those with mirror symmetry. For shapes typical of supercoiled DNA, writhe represents the number of supercoils formed by the axis of the molecule in 3D space, counted as the number of unreducible crossings of the axis averaged through all possible projections of the molecule. This number should not be confused with the number of solenoidal coils formed by the DNA axis. For example, for molecules wrapped n times around a cylinder with a pitch angle α_p, Wr = −n·sin(α_p) [67].

All three properties, Lk, Tw and Wr, can be expressed as line integrals along the ribbon and, therefore, are precise geometric properties. They are related via a simple exact relationship [65]:

Similarly, linking number deficit, ΔLk, partitions between torsional distortions and writhe:

Equation (10b) assumes that DNA is naturally straight and, therefore, Wr₀ = 0 when it is relaxed. Thus, formation of supercoils relieves torsional strain in DNA but restricts conformations in favor of the compact ones. ΔTw remains linearly proportional to the applied torque over a broad range of DNA deformations, both for over- and undertwisted DNA [26, 68–71]. Partitioning between twist and writhe is dictated by the balance of torsional and bending rigidities of DNA. In free DNA, about three quarters of linking number deficit are realized as writhe [72, 73].

The torsional rigidity determined from DNA stretching experiments, C = 4.3·10⁻¹⁹ erg·cm [68], tends to be somewhat higher than that derived from the analysis of DNA conformations, 3.1·10⁻¹⁹ erg·cm [74, 75], although lower values of 3.0·10⁻¹⁹ erg·cm and 3.5·10⁻¹⁹ erg·cm have also been reported [76, 77]. The reasons for this discrepancy are not well understood, but could reflect stress-induced changes in DNA mechanical properties. Alternatively, they could be related to changes in DNA conformational statistics imposed by bead attachment, as has been found previously for bending fluctuations of DNA [54].

Analogous to DNA bending fluctuations, the torsional rigidity is sometimes reported in terms of the torsional persistence length, P_t. This value describes how quickly the correlation between the twist of individual base pairs decays along the DNA length (see also Equation 7). Torsional persistence length is related to torsional rigidity by the equation: P_t = 2C/kT, where kT is the Boltzmann factor. Using the value 3.1·10⁻¹⁹ erg·cm determined from bulk measurements, one obtains a DNA torsional persistence length of 150 nm.

When DNA is twisted using magnetic tweezers, the results are markedly affected by the strength of the stretching force. At low forces, the introduced supercoils induce extrusion of a plectoneme, which contracts the DNA (Figure 6A). This results in characteristically symmetric “hat curves” when DNA extension is plotted against rotation (Figure 6B). For relatively short DNAs, one should expect only one plectoneme within a molecule. This is because the highest DNA curvature (and, hence, the highest energetic penalty for formation) is concentrated in the terminal loop of the plectoneme. It would be energetically unfavorable, therefore, if additional supercoils were initiating new plectonemes instead of being absorbed into the existing one. Under physiological conditions, long supercoiled DNAs produce multiple plectonemes, with the length of each branch averaging about 1.7 kb [72, 73]. A magnetic tweezers study of fluorescently stained DNA [78] revealed that plectonemes are highly dynamic at low stretching forces but become less mobile and less frequent as the force increases. Plectoneme formation is also facilitated by a reduction in the salt concentration, which leads to increased electrostatic repulsion between DNA segments [79].

The slope of the hat curves, about 40 nm per turn, can significantly increase at low stretching force [80, 81]. The critical level of DNA supercoiling required to extrude the plectoneme increases with increasing tension. This occurs because the formation of a supercoil reduces the torsional strain in the molecule but also must perform work against the stretching force. The balance of the two energies becomes favorable only above certain level of DNA superhelicity, and this critical level of σ is proportional to the applied force. A more detailed analysis of this phenomenon can be found in Strick et al. [82].

The symmetry of the hat curve disappears at forces above 0.4 pN due to supercoiling-induced structural transitions in DNA. At intermediate forces, these transitions are limited to negatively supercoiled DNA, resulting in asymmetric hat curves. When force exceeds 3 pN, positively supercoiled DNA also undergoes a transition to adopt the P-form, similar to Pauling's early proposal of the DNA structure, where bases are exposed to solution [83].

Notably, these transitions occur at lower levels of DNA supercoiling than in bulk experiments. The main reason for it is the suppression of writhing in stretched DNA. As a result, most of the supercoiling in stretched DNA is absorbed as twist, with very low contribution from writhe. By contrast, the linking number deficit is split 3:1 between writhe and untwisting under native conditions [84–86]. Thus, the same level of torsional strain that triggers structural transitions, would be achieved at 4-fold lower levels of supercoiling when DNA is stretched. From the comparison of the stretching curves for positively and negatively supercoiled DNA, the energy of the transition upon DNA untwisting was estimated as 4 kJ mol⁻¹ bp⁻¹ [82]. This is on par with other estimates for the formation of non-canonical DNA structures that were obtained in the absence of stretching forces [87–89].

Magnetic tweezers offer a convenient tool for handling two or more DNAs. This can be readily achieved by increasing the amount of DNA during its attachment to magnetic beads. Such beads end up tethered to the surface via multiple DNAs. Beads with two DNAs have been used for studies of DNA braiding [80, 90] and various enzymes that require proximity of distant DNA fragments, including DNA topoisomerases [91], condensins and transcription factors [6, 7]. The advantage of this approach is that the DNAs can be brought into contact or pulled apart simply by rotating the magnets. The contact region can be created with either left- or right-handed chirality, as desired, and the length of the overlap can be varied by changing the number of intertwinings between the DNAs.

Beads with multiple DNAs can be recognized by their characteristic hat curves (Figures 6C,D), which remain bell-shaped even at high tension. Indeed, DNA contraction occurs in this case due to geometric restrictions imposed by the intertwining of different DNA strands rather than extrusion of supercoils and, therefore, is not inhibited by stretching force. A detailed analysis of the interplay between supercoil extrusion and DNA braiding can be found in Charvin et al. [90]. The number of attached molecules can be determined from the analysis of force-extension curves. Data-fitting yields a persistence length which is linearly proportional to the number of stretched DNAs. This immediately follows from Equation 7b and the realization that bending several molecules by the same angle requires proportionally more energy.

The beads with two DNAs can also be distinguished from those with more molecules by the analysis of their hat curves [7]. All such molecules are expected to produce a sharp spike at ±0.5 turns (Figure 6C) owing to significant DNA contraction caused by the first crossing of the strands. This spike is expected to be symmetric for beads with two but not three or more DNAs.

A common task in single molecule enzymology is to detect single steps of a given DNA motor protein and determine their size. This task is often exacerbated by high noise in the observed DNA extensions. This high noise becomes especially problematic at low forces, when thermal fluctuations are prominent. The use of smaller magnetic beads and shorter DNAs helps partially alleviate this problem [41]. However, much work needs to be done at the signal processing level. Several automatic algorithms have been developed [38, 92–94] that allow hands-free deconvolution of reaction time courses into series of distinct steps, and new algorithms are likely to appear in the future. All these algorithms, however, produce unambiguous results only when the signal is well separated from the noise. In borderline cases, the deconvoluted step sequences contain both false-positive and false-negative steps.

The developed algorithms seek to discard statistically insignificant steps while giving users the opportunity to vary the stringency of noise filtration. Statistical significance of the resulting steps can be evaluated, for example, using the Student's t-test [93, 94]. Such analysis calculates the probability that the two time series that precede and follow the putative step occur so by chance. Steps with a T-value below a preselected threshold are rejected. The T-value must be computed using the generalized least squares approach:

where s is the size of the putative step, σ² is the variance of the distribution of the measured DNA extensions, N₁ and N₂ are the numbers of time points before and after the step, and R is a measure of auto-correlation within the time series, x_i:

With this in mind, a computer program generally needs to be developed that identifies statistically significant steps within a time course. The steps are typically approximated using the Heaviside function, and the timing of the steps seeks to optimize the statistical significance of the approximation. This approach, however, does not yield a unique solution, because the time interval over which the signal is averaged in Equation (11) depends on whether or not a step has been recognized in the vicinity. As a result, the ultimate deconvolution of a signal into a step series might vary depending on the algorithm.

Several step recognition algorithms have been described in the literature. All of them act iteratively by recognizing steps one at a time. The algorithm in Kerssemakers et al. [92] begins by fitting the biggest step within the entire time course and then works its way down in ever shrinking time intervals. Another approach is to give preference to prospective steps that occur earlier during the time course while seeking to detect them independently of each other [93]. An example of step identification in a noisy signal is shown in Figure 7.

Magnetic tweezers create a nearly homogeneous field that changes on the sub-millimeter scale, much larger than the specimens used in most studies. This offers an opportunity to handle several beads at once and thereby accelerate the collection of the required statistics. This is a significant advantage since accumulating enough statistics is time-consuming. To be effective, however, the multiplex approach must overcome three challenges [95–97]. First, all beads must be manipulated with a known force. Therefore, the force applied to each bead must be individually measured during calibration to account for the polydispersity in the beads' magnetic properties. Second, the computer program must be able to manage a high data acquisition rate: indeed, the computer must be able to process images of multiple beads and convert them into values of DNA extensions with adequate speed. Third, to increase the number of beads within the field of view, the magnification must be decreased without sacrificing image resolution too much.

To increase the density of beads, a method for targeted DNA tethering has been developed (Figure 8). In this method, a negative stamp of the preferred pattern is created on silicon wafer with electron beam lithography and dry etching [97] (I). Then, a flat elastomeric PDMS (polydimethylsiloxane) stamp is coated with antidigoxigenin antibody on one side, and pressed against the silicon template. In the process, only the desired protein pattern will remain on the PDMS stamp, which is then printed onto a glass slide (II). The surface is then passivated, DIG-labeled DNA is attached to the printed surface (III), and finally the magnetic beads are attached [97]. Targeted DNA tethering helps increase the density of beads within the field of view, in a well-arranged fashion. This prevents bead clumping and allows one to obtain clear images, so that diffraction patterns of numerous beads will remain separated during bead motion.

Using magnetic tweezers is a real time technique. Therefore, the sequence and duration of DNA twisting and stretching is tailored to specific experiments. However, several questions must be addressed at the design stage. The first and foremost question to be addressed is the choice of the substrate. The first experiments employed arbitrarily picked double stranded DNAs that were terminally labeled with derivatized nucleotides [14, 80]. Such molecules are easy to make and often suffice. In other cases, however, more elaborate structures are needed. Below, we describe several such structures that were used in magnetic tweezer experiments, including DNA with bulges and hairpins, single stranded DNA, RNA, and fluorescently labeled DNA.

The second critical question is whether or not supercoiling will be required. In order to observe supercoiling, one must use DNA with multiply labeled extremities. Such DNA could be cumbersome to produce, especially if the central part of the molecule itself includes multiple elements. If no DNA twisting is needed, however, more efficient labeling strategies can be used, such as filling sticky ends of the molecule via primer extension or extension of the ends by terminal transferase.

Improving the signal-to-noise ratio is a recurrent issue in single molecule experiments. In general, thermal noise in the system can be reduced by using shorter DNA, higher forces and increasing the number of observations. Thus, researchers should select the shortest DNA that would still support the studied reaction. The use of higher forces, however, could be limited by the internal constraints of the system. For example, motors that make large steps often exert only a weak force [6, 93], simply because the work performed by biological motors is often linked to the energy of ATP hydrolysis, ΔG_ATP ≥ F·s. Moreover, strong forces can induce undesired structural transitions in the system. Stretching DNA by the application of 30 pN force, for example, can lead to DNA unzipping or disruption of protein-DNA interactions.

Similarly, the number of observations can only be productively increased until they remain uncorrelated (Equation 11). At low frequencies, they are limited by the rate of the studied process and are outside of the observer's control. Thus, the time period over which the signal is averaged must not exceed the time between successive steps of the motor in question. At high frequencies, they are limited by the viscosity of the system, which sets the cutoff frequency of the bead's Brownian motion (Equation 2). Collecting data points at faster rates does not improve the signal-to-noise ratio. The cutoff frequency can be somewhat increased by switching to smaller beads owing to their reduced friction. Smaller beads, however, generate lower forces, which limits the experiment in its own way.

Depending on the task, various DNAs might need to be prepared. We review below the preparation of several such molecules. For proteins that do not require specific DNA sequences, such as DNA topoisomerases [91, 98] or condensins [6, 7, 93], plasmid DNA or PCR generated dsDNA, multiply labeled with biotin and digoxigenin at the extremities, are often used as target DNA substrates. Instead of sequence specificity, the binding of such proteins often requires DNA supercoiling or braiding, both of which can be achieved by rotation of magnetic beads. Multiple end labeling promotes stronger binding to the surface, and thus a longer-lived twisted state can be attained.

For proteins that require a specific nucleotide sequence, such as RNA polymerase (RNAP) or some DNA looping proteins, more complicated substrates have been incorporated in the experimental design. A DNA looping restriction endonuclease BspMI recognizes the asymmetric sequence 5′-ACCTGC-3′. λ-phage DNA, where this binding sequence occurs 41 times, has been used as a substrate for a magnetic tweezers experiment [99]. High affinity tandem repeats of 601, a nucleosome positioning sequence, helps to localize histones on DNA to form a compact nucleosome. These 601 sequences are convenient for the study of histones if repeatability of protein assembly on DNA is required [2, 3].

For experiments that require single stranded DNA, the substrate can be prepared by thermal denaturation of labeled dsDNA precursors [100] (Figure 9A). In this study, the dsDNA precursor has been heat treated (5 min at 100°C) shortly before injection into the flow chamber, and then quickly chilled and diluted into low salt, ice cold solution. To monitor DNA relaxation by type IA topoisomerase, a bulge of 25 nucleotides has been introduced into a plasmid DNA [101]. The resulting molecule was then digested with restriction endonucleases to generate a linear DNA fragment. The ends of this DNA fragment with the bulge were then multiply tagged with biotin and digoxigenin (Figure 9B).

Production of labeled RNA molecules has been achieved with the help of in vitro transcription [52, 81]. To this end, dsDNA templates were constructed that give rise to complementary RNA strands with unique leader sequences to facilitate further cloning. Following transcription, the two RNA strands were hybridized to generate dsRNA, and the unique ends have been used to attach the molecule to biotin- and digoxegenin-labeled RNA handles [52]. In another approach [26], terminally labeled ssRNAs were generated using a two-step transcription reaction. In the first step, an incomplete mixture of labeled nucleotides (which included biotin- or digoxigenin- derivatized adenosine or uracil triphosphate, as appropriate) was used to generate short, multiply labeled leader fragments. Such reactions stall once the need for the missing nucleotide arises. Thereafter, the labeled nucleotides were replaced with the full complement of the unlabeled ones to generate the body of the molecule (Figure 9C).

Hairpin DNA substrates have been used for studies of DNA unwinding by DNA helicases [30, 94]. These enzymes bind at the junction between single and double stranded DNA. Several approaches were described for the preparation of DNA hairpins. One such method involves primer extension of a 5′-biotin tailed primer annealed to a circular ssDNA using a DNA polymerase. The resulting circular product were linearized by restriction digestion. A short oligo was then ligated between two DNA strands to form a hairpin structure [30].

Another strategy for the preparation of a DNA hairpin is depicted in Figure 9D where biotin and digoxigenin anchoring ends and a pair of FRET (Főrster resonance energy transfer) dyes have been combined into one molecule [102]. This kind of substrate is particularly useful in cases where both force measurement and fluorescent tracking are desirable, such as tracking the movement of DNA motor proteins under force. Here, the DNA hairpin was prepared by combining two separate DNA substrates. The first substrate was made by annealing two matching oligonucleotides, which contain a sticky end, biotin and the cyanine fluorophores (Cy3 and Cy5) on one end and a DNA loop at the other (Figure 9D, I). The second substrate is a DNA spacer with a matching sticky end and digoxigenin-labeled DNA handle (Figure 9D, IV). The two substrates were then ligated together via their matching sticky ends, producing a structure with DIG and biotin on opposite ends (Figure 9D, V).

Another special type of substrate used in magnetic tweezers experiment is the Holliday junction [102]. This structure was created by annealing four oligonucleotides which contain four mutually complementary single stranded regions with biotin and the FRET pair located at the appropriate arms. A bacteriophage DNA spacer with a digoxigenin-labeled DNA handle was attached to the four-arm Holliday junction (Figure 9E).

Magnetic tweezers are an excellent tool for probing mechanical properties of DNA including: DNA unwinding [103] bending [5] and elasticity [47], activity of DNA topoisomerases [91, 98], DNA condensation [6, 7, 93], and functions of chromatin remodeling enzymes, DNA helicases and translocases [30, 94, 104], RNA polymerase [103, 105], restriction endonucleases [99], and many others. Below, we review several applications of magnetic tweezers that highlight their power and versatility in studies of DNA modifying enzymes. Note that many features uncovered in these studies would be very difficult to reveal using the more traditional biochemical approaches.

Unraveling complex reactions can often benefit from the use of multiple probes. In this respect, simultaneous detection of single molecule fluorescence and deformation offers new possibilities to mechanistic studies. An example of this has been recently reported using DNA hairpin formation for the proof of concept [102].

To observe the event, Kemmerich and coauthors constructed a molecule that contained a self-complementary region of single-stranded DNA with a FRET pair at its borders and DNA handles as needed for stretching (see Figure 9 for details of the construction). Without applied force, the single-stranded region folds into a hairpin, thus bringing the two fluorophores together (Figure 11B). The high-FRET state can then be detected simultaneously with DNA contraction, which accompanies formation of the hairpin. Conversely, an increase in the force melts the hairpin and induces a low-FRET state in the system.

In another example, Strick and coauthors combined magnetic trapping with total internal reflection fluorescence, TIRF, [111–113] to elucidate the mechanism of bacterial transcription-coupled DNA repair (Figure 11C). The use of evanescent light helped detect fluorescence from the tagged proteins while eliminating at the same time the interference from the bead. Magnetic force and fluorescence have been used to provide complementary information, which helped decode a complex sequence of events that unfold during the interaction between RNAP and the Mfd translocase during their encounter with DNA lesions. Specifically, the catalytic properties of RNAP were probed using stretching force, whereas fluorescence was reporting on the composition of the complex.

Magnetic tweezers are most frequently used for studies of DNA and DNA reconfiguring enzymes. However, a number of other applications have been recently described. These include studies of protein folding [5, 114], rotating protein assemblies [115], cellular cortical stiffness [9], cellular mechanotransduction processes [10] and intracellular transport [11]. These applications take advantage of the non-invasive character of the tweezers. Indeed, magnetic tweezers do not require intense sample irradiation, efficiently work in solution, and do not even need a computer for some tasks. We will review in this section several of the applications that do not focus on DNA.

Magnetic tweezers are powerful instruments well suited for diverse single molecule applications. They are lauded for their ability to twist macromolecules in addition to stretching. However, they have a number of other unique advantages. In particular, they provide a technical solution that does not use intense irradiation of the sample, which inevitably leads to accelerated degradation. Furthermore, the tweezers generate a nearly homogeneous force field over large distances. As a result, no adjustments are needed during the course of experiment to compensate for enzyme translocation. Additionally, the flow chamber can be readily modified for multiplex applications so that many reactions can be followed in parallel. Finally, magnetic tweezers can handle multiple DNAs as easily as they handle single molecules.

Recent advances in hybrid single molecule techniques, which combine magnetic tweezers with microfluidic systems and fluorescence microscopy, opened new possibilities of experimental design. These advances also helped to solve one of the original weaknesses of the method, its difficulties in generating strong magnetic fields. The use of micromanufactured magnetic poles allows a significant reduction in the separation between the bead and the pole, which helps produce stronger field gradients and stiffer magnetic traps. Thanks to such developments, sub-nanometer resolution has become possible.

The method has few limitations. One of them includes difficulties in working with particles that are susceptible to magnetic fields. Among biomolecules, however, the list of such molecules is rather short and consists mostly of proteins associated with metal clusters. Perhaps the biggest limitation is the rather bulky geometry of the magnetic poles, which must be positioned close to the sample. As a result, combining tweezers with other applications requires a degree of ingenuity. Likewise, carrying beads around using magnetic tweezers remains a challenge and requires a specialized instrument. These limitations, however, are not especially restrictive and only underscore how versatile and powerful the technique truly is.

RS and VR designed and wrote this manuscript, approved its final version and are responsible for its content.

This work was supported by the awards for the project number HR14-042 from Oklahoma Center for Advancement of Science and Technology, and HDTRA1-14-1-0019 from the Department of the Defense, Defense Threat Reduction Agency.