In this section, the fundamental properties of single DNA chains related to polymer physics are introduced. For experiments in polymer physics, monodisperse polymers should be used to obtain precise measurements. Several polymerization methods, known as living polymerizations, can provide monodisperse polymers with a very narrow distribution of molecular weight or degree of polymerization (DP), ideally a Poisson distribution. A DP distribution cannot be eliminated from usual polymer syntheses. In contrast, genetically prepared DNA molecules from viruses or bacteria have uniform length for each species, so DNA molecules in solution are suitable for polymer physics experiments. In the first half of this section, the fundamental properties of single DNA chains determined from simple direct imaging are described. Some applications using optical tweezers are provided in Sect. 14.3.

To reveal the dynamics of single polymer chains in solution, the diffusion coefficient D, radius of gyration R _g, and persistent length P are essential physical quantities. These polymer solution properties have been studied mainly by ensemble measurements such as light and neutron scattering, viscoelasticity, dielectric relaxation, and electric birefringence. As described in Chap. 2, the rheological properties of polymer solutions are closely related to the conformation of polymer chains. The enormous effort spent on these characterization techniques has evolved into a successful molecular model (the Rouse-Zimm model) and scaling concepts. However, realistic images of the fluctuation of polymer chains have not been readily accessible to most researchers. Therefore, it was worthwhile to attempt the direct and real-time observation of single DNA molecules in solution.

D of a polymer molecule can be accurately determined from the mean square displacement against time (diffusivity), which is analyzed by tracking the center of mass of DNA fluorescence images. The hydrodynamic radius R _H of DNA molecules can be deduced from the Stokes-Einstein relation. Yanagida et al. compared their results with a theoretical model of wormlike chains [13], and P was determined to be 47–70 nm, in good agreement with the value measured by light scattering (~50 nm) [1].

The situation for the estimation of R _g is somewhat different. Fluorescence images of long DNA molecules such as bacteriophage T4 and λ show a thick filament with an apparent length of less than 10 μm without any disturbances (flow, electric field, osmotic pressure) because of the shortening of end-to-end distance reflected by random walk statistics. During fluctuation, the apparent length, shape, and local segment density (reflected by fluorescence intensity) of DNA fluorescence images change very quickly. R _g can be estimated by measuring both long- and short-axis lengths of sequential fluctuating DNA images. As a result, this estimation includes a substantial margin of error.

Matsumoto et al. analyzed the Brownian motion of T4 DNA and reported that R _g and rotational relaxation showed qualitative agreement with the Zimm model [5, 14]. However, because of the walls of the thin sample chambers, D deduced from diffusivity were somewhat lower than those predicted by the Zimm model. In 1996, Smith et al. clearly demonstrated that the power-law scaling of D with DNA length L is in good agreement with the Zimm model (D ~ L ^−3/5) using deeper chambers (depth of 75 μm) [6]. A scaling exponent ν of 0.611 ± 0.016 was reported. In a further detailed study, Robertson et al. reported a topology-independent scaling law of D ~ L ^−ν where ν _L = 0.571 ± 0.014, ν _C = 0.589 ± 0.018, and ν _S = 0.571 ± 0.057 for linear, relaxed circular, and supercoiled DNA, respectively [15]. The renormalization group theory that takes into account the effect of excluded volume predicts ν = 0.588 [16].

Ashkin first used optical tweezers for trapping micron-sized particles by laser radiation pressure [17]. Ashkin and his colleagues then expanded this technique to laser cooling to trap atoms. A prototype of current typical optical tweezers based on a single focused laser beam was also reported by Ashkin et al. [18]. They achieved optical trapping of 20-nm particles in water at room temperature and also manipulated viruses, bacteria, and organelles of protozoa by optical trapping [19, 20]. As an extension of these works, Chu and colleagues manipulated individual DNA chains using optical tweezers [21, 22].

The principle of optical tweezers to manipulate single DNA chains is illustrated schematically in Fig. 14.2 [23]. In typical setups using a single laser, an infrared laser beam is focused by an objective lens to trap beads attached to the end of DNA chains. Objectives with high numerical aperture (1.3 or higher for oil immersion) used for high-resolution fluorescence imaging can also generate a strong trapping force. The Nd:YAG laser emitting at 1,064 nm is the most widely used example at present. To manipulate a single DNA chain in the simple setup, in which a laser is focused in a fixed position, microfluidic chambers are combined with a microscope stage or a microscope stage is mechanically moved with submicron accuracy.

In early studies such as that of Matsumoto et al. [24], shear force was generated by slowly moving a cover slip to extend DNA chains to nearly their contour length. Optical tweezers are a sophisticated method to carry out similar experiments. Chu first demonstrated the elastic properties of single DNA chains using a pair of controllable optical tweezers to manipulate two beads attached at both ends of a DNA molecule via biotin-avidin bridges [21]. After the extension of λ-DNA to its contour length, the relaxation process to the random coiled state was captured after the release of one bead. The distance between the two beads decreased exponentially like a simple Hookean spring (Fig. 14.3).

Perkins et al. subsequently performed relaxation experiments of fully extended DNA molecules using a setup with a single laser and stopped-flow chamber [23]. Fitting these relaxation data with an inverse Laplace transform showed sharp spectral peaks, indicating that the relaxation process was a sum of exponentials (internal relaxation modes). The longest relaxation time scaled as L ^1.65, in agreement with the dynamic scaling prediction of the Zimm model. A detailed discussion of these findings is provided in Shaqfeh’s review [25].

Optical tweezers opened up a new research field called single-molecule DNA mechanics. Video microscopes equipped with dual-controllable optical tweezers or a single laser with a micropipette allows us to measure the force during DNA stretching. Similar experiments are widely carried out to investigate protein folding using atomic force microscopy (AFM) by AFM fishing. In contrast to AFM fishing for proteins, double-stranded DNA does not produce characteristic structures in its force-extension curves for full extension of the B form (normal double helix form) (Fig. 14.4), [83]. In 1992, Smith et al. reported the first quantitative force (F)-extension (x) curves for λ-DNA at various NaCl concentrations using magnetic bead trapping to measure the elastic responses of B-form DNA [10]. Bustamante et al. pointed out that fitting these data with the wormlike chain model gave better agreement than that with the freely jointed chain model [26]. Marko and Siggia suggested an analytical formula for F–x curves in terms of L and P:
where k _B is the Boltzmann constant and T is the absolute temperature [27]. Fitting F–x curves to this equation gives an intrinsic elastic persistent length of 50.8 nm. The overstretching region in F–x curves where normal double helix structure (B-form) is no longer retained is not a subject of this chapter, but modulations, unzipping, and melting of double helix structure are introduced in the review by Bustamante et al. [28].

Apart from dilute polymer solutions, the properties of concentrated polymer solutions are very complicated, but can be treated by the reptation model developed by de Gennes [29, 30] and Doi and Edwards [16]. The basic concept of this model is that a polymer chain in the entangled state is topologically constrained by the surrounding chains and relaxation processes are achieved by chain movement along its contour or inside a hypothetical tube. Perkins et al. directly observed the reptation movement of single DNA chains [22]. In their report, a fluorescently stained DNA chain embedded in background of unstained DNA chains was first extended by manipulating the bead attached to the end of the stained DNA molecule using optical tweezers, and then the extended DNA chain was retracted along the path previously drawn by the bead from the unfixed end and relaxed to its equilibrium state. The top and middle rows of Fig. 14.5 show that a small loop made by the dragged bead stayed fixed for long time (~120 s), indicating the constraint from the surrounding unstained chains was hardly modulated because of their slow diffusion. These observations provide direct evidence for the fundamental properties of the reptation model.

As a further detailed study, Smith et al. [31] determined the self-diffusion coefficient D _rep of λ-DNA in a concentrated region of a similar sample to that used in their previous work [22]. Their obtained scaling exponent α of 1.8 ± 0.1 for λ-DNA at high concentration was close to the scaling prediction for D _rep ~ L ^−α with α = 2. Reptational diffusion of λ-DNA was observed at 0.63 mg/mL, which is 16 times higher than the overlap concentration, but not at 0.40 mg/mL. In addition, shorter DNA with a concentration of 0.63 mg/mL did not follow the reptation model. These observations suggest that both chain length and concentration are factors that control the applicability of the reptation model to semidilute polymer solutions.

DNA gel electrophoresis is a widely used tool in biology and biochemistry. However, DNA migration modes in gels were not clarified for a long time. Before using single-molecule DNA imaging, linear dichroism, fluorescence polarization, and birefringence were employed to investigate DNA orientation in gels. However, the results obtained from these methods were difficult to interpret. Direct observation of fluorescently stained DNA migrating in gels can provide easily understandable descriptions concerning the molecular motion of DNA chains in gels. The behavior of DNA inside gels resembles that of entangled polymer solutions. First, the fundamental principles of DNA gel electrophoresis are briefly introduced.

In buffer solutions, DNA molecules migrate toward the positive electrode under a steady electric field. The migration velocity of DNA can be modulated by changing the ionic strength or applied electric field. However, the velocity is independent of DNA length when electrophoresis is performed in homogeneous buffer solution without a supporting gel [32]. Electrophoretic mobility μ, which is the ratio of migration velocity to applied electric field, can be written as the ratio of DNA effective charge Q to friction coefficient η. Both of these quantities are proportional to DNA length in homogeneous buffer solution, indicating that μ is independent of DNA length.

As described in Sect. 14.3, the relaxation of a single polymer chain in entangled polymer solution can be modeled by reptation movement. In the case of DNA electrophoresis, DNA chains are dragged inside gels by the electrophoretic force. The movement of DNA chains in gels under steady electric field can be explained by the biased reptation model, which takes the electrophoretic force into account. When DNA chains migrate inside a gel, the dependence of μ on DNA length can be divided into three regimes that are characterized by the relationship between gel concentration, electric field strength, and DNA length [33].

The first regime is the Ogston regime [34] that shows weak length dependency of μ on DNA length (low separation capability), because the gel pore size is larger than that of shorter DNA fragments. The second regime is called the reptation regime without DNA chain stretching (gels with small pores and weak electric field). In the second regime, μ clearly depends on DNA length. Numerous mathematical descriptions for the reptation mode in DNA electrophoresis have been developed [35–38].

The third regime is called the reptation regime with DNA chain stretching (gels with small pores, strong electric field, and longer DNA chains). In this regime, μ of longer DNA chains is almost independent of DNA length because of the strong entanglement between the gel network and DNA chains. Therefore, agarose gels possessing large pores are suitable to separate long DNA chains, whereas polyacrylamide gels with small pores can be used only to separate short DNA fragments (<1 kbp). These three regimes are also observed in electrophoresis of SDS-solubilized proteins [39] and synthetic polyelectrolytes [40]. Therefore, these phenomena are the general properties of linear polyelectrolytes.

According to the biased reptation model [37] applied to DNA electrophoresis, the velocity v of DNA inside a hypothetical tube can be expressed by the following equation:
where h is the length of projection of a mean end-to-end distance of DNA chain to the direction of the electric field, E is the applied electric field, and ξ is the frictional coefficient between DNA and the hypothetical tube [35]. This equation can be transformed to μ:
If the mean end-to-end distance of DNA under a weak electric field is same to that of coiled DNA without an electric field, indicating the square of h is proportional to L, μ is proportional to the inverse of L. This behavior is observed in the second regime of DNA electrophoresis in gels. Under a strong electric field, as long DNA chains are stretched along the electric field, h is proportional to L. Then, the dependence of μ on DNA length decreases again in the third regime. DNA chains longer than 50 kbp cannot be separated under common electrophoresis conditions with a constant electric field [41].

The separation of long DNA chains (>50 kbp) using pulsed-field gel electrophoresis (PFGE) [42, 43] was the outstanding breakthrough in the third regime of DNA gel electrophoresis. In a PFGE chamber, the electric field is periodically altered between two electrode pairs with a fixed angle from 90° to 120°. PFGE allows DNA chains up to 10 Mbp in length to be separated. A molecular sieving effect is thought be generated because the switching time required for the direction change of DNA migration depends on DNA length.

To assess the above models, Smith et al. employed fluorescence video microscopy to observe the migration of DNA on gels under both steady and pulsed electric fields [44]. Under a steady electric field, DNA chains in the gel migrated in an indirect manner with repeated extension and contraction movements. With increasing electric field strength, DNA chains became stretched and better aligned with the electric field. Another configuration, “U-shaped” DNA chains that caught on obstacles in the gel, was often observed. After stretching to its contour length, the shorter arm of a DNA chain slipped into the tube made by the longer arm, indicating the DNA chain became free from the obstacle. Schwartz and Koval also reported similar observations [45].

In PFGE, the slipping motion of DNA chains is modulated in different directions by placing two electrode pairs with a fixed angle from 90° to 120° between them. Smith et al. reported that tugged DNA segments (tails) changed to leading segments (head) corresponding to changes in the electric field direction with an electrode angle of 120°, just like a train running on a switchback track [44]. These reorientation dynamics are clearly shown in Fig. 14.6 [46]. Their computer simulations demonstrated that shorter DNA chains change direction more quickly and migrate faster than longer chains. Longer chains take a longer time to backtrack on their previous paths than shorter ones. When a 90° angle between electrodes is used in PFGE experiments, both ends can become the leading end, resulting in poor size separation.

The term DNA condensation is reserved for the conformational change of elongated DNA molecules into collapsed states with ordered morphologies, typically toroidal and rodlike states [47]. DNA condensation is a characteristic phenomenon related to a traditional problem of polymer physics, the coil-globule transition. Transmission electron microscopy, light scattering, and sedimentation have been used frequently to characterize DNA condensation. However, using these techniques, researchers could not easily determine whether DNA condensation is induced at single-molecule level or involves aggregation of multiple chains. Therefore, the definition of DNA condensation does not discriminate between single-molecule collapse and aggregation, although the former was predicted theoretically [48]. In concert with this prediction, Widom and Baldwin concluded that (1) the monomolecular condensation of λ-DNA induced by cobalt hexammine is not a two-state reaction and (2) the transition for monomolecular condensation is diffuse [49]. As described above, a consensus view for DNA condensation has not yet been provided. Thus, the direct, real-time observation of single DNA molecules may allow better characterization of DNA condensation.

DNA condensation is induced by neutralizing the negative charges of phosphate groups under various conditions. Multivalent cations (charge of 3+ or greater in normal aqueous solution), cationic surfactants, cationic lipids, and cationic polymers are well-known DNA condensing agents. Neutral polymers that exert a crowding effect through an excluded volume mechanism can also induce DNA condensation in the presence of adequate concentrations of salts (typically psi-condensation using poly(ethylene glycol) (PEG) and Na⁺) [50]. Light scattering studies revealed that the intensity of scattered light from DNA solution increases continuously with the addition of condensing agents [51]. However, the scattering curves of DNA condensation cannot be used to determine how DNA molecules collapse into condensed states, particularly if it is in a continuous or discrete manner. Therefore, physicochemical characterization techniques that clarify the detailed condensation behavior of DNA are desired.

Yoshikawa and colleagues used fluorescence microscopy to reveal the transition behavior of DNA condensation and found that both the elongated and collapsed states coexisted during the transition region of DNA condensation induced by polyarginine [52]. Intensive study by the Yoshikawa group elucidated that this biphasic behavior of DNA condensation is a common feature independent of condensing agents. Mel’nikov et al. reported the relationship between the biphasic distribution of T4 DNA conformation and cooperative binding in DNA condensation induced by the cationic surfactant cetyltrimethylammonium bromide (CTAB) [53]. In this system, the binding of CTA ions onto DNA was quite similar to the aggregation of surfactant molecules observed in micelle formation in the closed association model. Until DNA condensation was complete, the bound fraction of CTA ions increased linearly with the concentration of CTAB, while the concentration of unbound CTA ions remained consistently lower. Takahashi et al. reported that the width of the coexistence region narrowed as the valence of polyamines increased [54]. Increasing the valence of the polyamine by one corresponded to a decrease of the transition point of DNA condensation, expressed as the concentration of added condensing agent required for the transition, by one order of magnitude. This experimental trend agreed with a theoretical prediction made using the modified Flory-Huggins theory taking into account the electrostatic interaction between phosphate groups of DNA and polyamines.

Yoshikawa and Matsuzawa used single-molecule DNA observation to visualize how T4 DNA chains collapse into condensed states in the case of psi-condensation using PEG and NaCl [55]. Because the PEG concentration required for DNA condensation increases the viscosity of the sample solution [50], the fluctuation of DNA chains slowed as the DNA molecules became entangled in gel matrices, which can easily be observed by a fluorescence microscope. During the condensation process of this system, nucleation occurred most frequently at the end of the DNA chains and the remaining elongated part successively wound around the nucleation center within 10 s, i.e., the nucleation and growth processes were confirmed as shown in Fig. 14.7 [55]. These studies reveal that the continuous increase of scattering intensity observed in light scattering experiments of DNA condensation can be explained by the gradual change in the distributions of both elongated and collapsed states in the transition region of DNA condensation that is characterized by the first-order phase transition of single DNA chains. In addition, Yamasaki and Yoshikawa demonstrated that DNA condensates collapsed by Fe³⁺ ions unfolded to the elongated state upon addition of ascorbic acid solution, which reduced Fe³⁺ to Fe²⁺ [56]. This indicates that the collapsed and elongated conformations of DNA can be controlled by the electrochemical redox reactions of condensing agents.