The Statistical Segment Length of DNA: Opportunities for Biomechanical Modeling in Polymer Physics and Next-Generation Genomics

Abstract

The development of bright bisintercalating dyes for deoxyribonucleic acid (DNA) in the 1990s, most notably YOYO-1, revolutionized the field of polymer physics in the ensuing years. These dyes, in conjunction with modern molecular biology techniques, permit the facile observation of polymer dynamics via fluorescence microscopy and thus direct tests of different theories of polymer dynamics. At the same time, they have played a key role in advancing an emerging next-generation method known as genome mapping in nanochannels. The effect of intercalation on the bending energy of DNA as embodied by a change in its statistical segment length (or, alternatively, its persistence length) has been the subject of significant controversy. The precise value of the statistical segment length is critical for the proper interpretation of polymer physics experiments and controls the phenomena underlying the aforementioned genomics technology. In this perspective, we briefly review the model of DNA as a wormlike chain and a trio of methods (light scattering, optical or magnetic tweezers, and atomic force microscopy (AFM)) that have been used to determine the statistical segment length of DNA. We then outline the disagreement in the literature over the role of bisintercalation on the bending energy of DNA, and how a multiscale biomechanical approach could provide an important model for this scientifically and technologically relevant problem.

1. Introduction

The configurational statistics of a polymer chain are described by its contour length, L, statistical segment length, b, and effective width of the polymer backbone, w. The ratio of the first two quantities is a way to express the degree of polymerization in terms of the number of statistical segment lengths, N = L/b, while the ratio of the second two quantities, b/w, provides a measure of the flexibility of the chain. In the limit N ≫ 1, the configurations (and other properties) of polymers approach universal limits that are independent of the underlying chemistry. This is not to say that the chemistry becomes irrelevant at larger length scales. Rather, the chemical details are embedded in the statistical segment length. For example, the statistical segment length of polyethylene (a linear hydrocarbon chain) is 0.59 nm, while the bulky side groups of a polystyrene chain increase this value to 0.67 nm [1]. These synthetic polymers are flexible chains, where b ≈ w, and their underlying chemistry is relatively simple.

Biopolymers tend to be very stiff chains (b/w ≫ 1) and often have much larger statistical segment lengths than polyolefins. We will consider here the case of double-stranded deoxyribonucleic acid (DNA), whose statistical segment length in a high ionic strength buffer (around 100 mM) is of the order of 100 nm [2]. In many physical experiments and engineering-driven work, the DNA is often labeled with an intercalating dye such as YOYO-1. These molecules are essentially nonfluorescent in solution, but when inserted into the DNA backbone, their fluorescence increases by almost 1000-fold [3]. How the intercalation process affects the statistical segment length of DNA remains a controversial question, with conflicting experimental data [4–11]. Quantifying the role of intercalation on the statistical segment length would have widespread impact on a number of problems in polymer physics and genomics technology.

From the polymer physics side, the pioneering work by Chu and coworkers [4,12,13] lead to the widespread use of DNA as a model system for visualizing the dynamics of polymers in flow. These experiments rely on intercalating dyes to visualize the DNA, and their interpretation requires understanding the role of these dyes on the mechanical properties of DNA that are embedded in the statistical segment length. Of particular interest is the case where a small number of fluorescently labeled DNA molecules are immersed in a sea of unlabeled molecules. This method has been used to understand, among other things, the deformation of DNA as it is pulled under an externally applied force through a polymer solution [14], the dynamics of DNA in a contraction flow [15], and the possible mechanisms underlying shear banding (or lack thereof) in entangled polymer solutions [16], the latter problem being arguably the most controversial question in polymer rheology at the present time. These experiments have been enabled by accompanying advances in microfluidics that permit the fabrication of precise geometries for studying such flows [17]. The fluorescently labeled DNA molecules are intended to serve as “test chains” whose dynamics can be tracked by microscopy, thereby providing a direct test of dynamical models of polymer flow [18,19]. Implicit in this experiment is the assumption that the test chains have the same mechanical properties as the background chains. If not, then the interpretation of the experiments becomes more complicated. While the experiments cited so far (and the discussion that follows in later sections) focus on double-stranded DNA stained with a bisintercalating dye such as YOYO-1, there is an emerging body of literature due to Schroeder and coworkers that focuses on dyed single-stranded DNA as the model polymer, both in linear [20] and branched [21] topologies. A similar question applies to all fluorescent labeling methods—do the dyes affect the mechanical properties of DNA?

My interest in understanding the role of intercalation is connected to a next-generation genomics technology known as “genome mapping in nanochannels” [22,23], which provides an important complement to DNA sequencing for de novo assembly [24,25] and the identification of structural variations in genomes [22,26]. In this technology, illustrated in Fig. 1, DNA with sequence-specific labels are stretched by confinement in a nanochannel to read off the physical distance between the labels by fluorescence microscopy. If the fractional extension of the DNA in the nanochannel is known, for example by calibrating the experiment with a known molecular weight standard, one can convert the physical distance between labels (in nm) to a genomic distance (in base pairs). Comparing the resulting data to the reference genome allows one to determine the location and nature of large-scale rearrangements in the DNA [27,28]. Alternatively, one can assemble the single molecule data into a scaffold for de novo genome sequencing.

Fig. 1.

Schematic illustration of genome mapping in nanochannels. Genomic DNA is labeled with fluorescent markers by nick labeling and repair [22] to insert markers at the nick site location (bolded sequence). The DNA are injected into a nanochannel and imaged by fluorescence microscopy. The resulting fluorescence intensity image can be digitized by a threshold filter to provide the physical distance between nicking sites in the nanochannel. Through knowledge of the fractional extension of the confined chain, the physical distance (in nm) can be converted to a genomic distance (in base pairs), producing a “barcode” for that DNA molecule. Repeating the process with many molecules from the same individual, and then assembling their overlapping parts, produces the genomic map. In this particular schematic example, the genomic map is used to identify an insertion, where additional DNA has been inserted between two of the nick sites. This is just one of many different types of structural variations [27] that can be readily identified by genome mapping in nanochannels. Modified from Ref. [28] with kind permission of the European Physical Journal (EPJ).

The commercial system for performing these experiments uses circa 45 nm nanochannels [22], and simulations and theory [29–32] show that the properties of DNA confined in such a small channel are very sensitive to the statistical segment length (or, equivalently, the persistence length). Explicitly, Odijk [29] proposed that the key physical parameter governing the extension of DNA in a nanochannel is the so-called global persistence length g, which quantifies the typical distance between hairpin turns [33]. We have computed the global persistence length as a function of D/ℓ_p, where D is the channel size, for multiple channel geometries by simulating the discrete wormlike chain model described in Sec. 2 [32,34,35]. Similar results were obtained recently by Chen as well, using a powerful propagator method to compute g in circular tubes [36]. As predicted by Odijk [33], the global persistence length has an exponential dependence on D/ℓ_p, albeit not exactly of the form proposed by his theory [32,36]. For example, in a D = 45 nm wide, square nanochannel, a persistence length of ℓ_p = 50 nm leads to a global persistence length g = 420 nm using the result of Muralidhar et al. [32]. However, a modest increase in the persistence length to ℓ_p = 55 nm or 60 nm leads to global persistence lengths of 650 nm and 990 nm, respectively. In other words, a 10% or 20% difference in persistence length, which is well within the experimental range that we will see below, leads to a 54% or 134% increase in the global persistence length. As such, obtaining an accurate measure of the DNA persistence length is critical to modeling its conformation in confinement.

In addition to its fundamental importance in the theory of channel-confined DNA, the global persistence length has important practical implications for genomic mapping. In the experiment, the DNA is imaged for approximately 150 ms to obtain the position of the fluorescent markers on its backbone. The variance in the measured DNA extension is, to a first approximation, equal to the product of the distance between the markers and the global persistence length [29,32] for channels D ≈ b. Obtaining accurate measurements of the physical distance between the markers requires that this variance be small. The predictions of Odijk's theory [29] and accompanying simulation efforts [31,32] agree with experience. Genome mapping in channels near the statistical segment length is challenging [37], as these channel sizes are coincident with the maximum in the variance in DNA extension with respect to channel size [31,32]. However, the accuracy improves substantially in smaller channels [22] because the global persistence length becomes so long that hairpin turns are essentially suppressed [29,33]. Thus, the success of this genomics method relies on an accurate understanding of the statistical segment length of dyed DNA.

The overarching goal here is to highlight how biomechanical modeling could advance our understanding of the role of intercalating dyes on the statistical segment length of DNA. This is a source of uncontrolled systematic error in both the polymer rheology and genomics areas. Within the polymer rheology community, there is little discussion of the role of intercalating dye on experimental results. Within the genomics community, the problem receives more discussion but little resolution. For example, simulations and theory for moderately confined DNA (the so-called “extended de Gennes” regime [30]) now agree with experimental data to within about 10% [38–40]. While there are several sources of potential error [40], the unknown effect of the intercalating dye remains on the list of suspects. Unfortunately, it seems that the communities that most urgently need to answer this question are the ones least likely to solve it. As a result, understanding the role of intercalating dyes, in particular via multiscale modeling approaches, offers enticing opportunities for the biomechanical community to impact multiple related research fields.

With this motivation in mind, this perspective aims to provide researchers in the biomechanical modeling community with the requisite background to understand the polymer physics problem, assuming little previous knowledge of the area. We begin by reviewing the wormlike chain model for DNA, and illustrate the connection between the persistence length and the statistical segment length. We then describe three standard methods for measuring the statistical segment length (light scattering, optical and magnetic tweezers, and atomic force microscopy, AFM), with a focus on their capabilities and limitations to address the role of intercalation. These two sections are intended as a pedagogical introduction to the relevant polymer physics for readers who are more familiar with biomechanical problems. We then outline the ongoing controversy over the role of intercalating dyes on the statistical segment length of DNA, and suggest how the particular skills of the biomechanics community, most notably the ability to construct models across multiple length scales, could be brought to bear on this problem.

2. Wormlike Chain Model for DNA

DNA is generally regarded as a wormlike chain [41]. This model regards the polymer as the continuous curve r(s) illustrated in Fig. 2(a), where s is the arclength and r is dimensionless. The local radius of curvature is the inverse of $|\partial \mathbf{r}/\partial s|$. The bending energy for a configuration r(s) is then obtained by summing up the inverse-squared radius of curvature over the entire chain [41]

Fig. 2.

Illustration of the wormlike chain model. (a) Continuous wormlike chain parameterized by the curve r(s). The bending energy is defined locally by Eq. (1) based on the curvature of the backbone. (b) Discrete wormlike chain model consisting of bonds of length ℓ. The relevant angle for a given bend is given by the dot product between the vectors r_i quantifying the orientation of neighboring segments. (c) Plot of the discrete wormlike chain bending energy given by Eq. (6).

$$\frac{E_{\mathrm{b}}}{k_{\mathrm{B}}T}=\frac{{\ell }_{\mathrm{p}}}{2}{\int }_0^L{\left|\frac{\partial \mathbf{r}}{\partial s}\right|}^2\hspace{0.17em}\mathrm{d}s$$ (1)

where the persistence length ℓ_p arises in the formalism as the characteristic length over which the chain can bend under the thermal energy k_BT.

In 1949, Kratky and Porod obtained a result for the root-mean-squared end-to-end distance of a continuous wormlike chain [42]

$$R_{\mathrm{e}}^2=2{\ell }_{\mathrm{p}}L-2{\ell }_{\mathrm{p}}^2\left[1-\text{exp}\left(-\frac{L}{{\ell }_{\mathrm{p}}}\right)\right]$$ (2)

In the limit of a long chain, L ≫ ℓ_p, Eq. (2) reduces to

$$R_{\mathrm{e}}=\sqrt{2{\ell }_{\mathrm{p}}L}\hspace{1em}(L\gg {\ell }_{\mathrm{p}})$$ (3)

In the context of DNA (and other semiflexible chains), the persistence length and the statistical segment length are often used interchangeably via the relationship

$$b=2{\ell }_{\mathrm{p}}$$ (4)

To see how this result arises, we note that the end-to-end distance of an ideal chain obeys Gaussian statistics [43]

$$R_{\mathrm{e}}=b{N}^{1/2}=b\sqrt{\frac{L}{b}}=\sqrt{\mathit{L}\mathit{b}}$$ (5)

where the statistical segment length plays the role of the step size in the random walk. Comparing Eqs. (3) and (5), we recover Eq. (4) in the limit L ≫ ℓ_p. Note that this result is only valid for wormlike chains; flexible chains such as polyolefins tend to have persistence lengths that are similar to their statistical segments because the backbone correlations hardly “persist” for such polymers [43].

It is worthwhile to keep in mind that the statistical segment length and the end-to-end distance are thermodynamic quantities that can be obtained from equilibrium statistical mechanics. As such, they cannot depend on hydrodynamic interactions or any dynamic phenomena. The two concepts are connected, as the configurational phase space of the polymer determines whether it is in the Rouse limit [44], where hydrodynamic interactions are negligible, or the Zimm limit [45], where hydrodynamic interactions are substantial, or somewhere in between [2]. For example, the hydrodynamics of a rod-like polymer in the limit L ≪ ℓ_p in Eq. (2) are quite different from those for the coiled conformation in the opposite limit embodied by Eq. (3). However, for a single polymer chain at equilibrium in a quiescent solution, it is the thermodynamics that dictate the configurations, which then give rise to the hydrodynamics. Indeed, the primacy of the thermodynamics is the key to a number of approaches for approximating the diffusivity of the chain, such as the widely used Kirkwood–Riseman approximation [46,47], and the reason why the fluid viscosity cannot appear in the formulation of the present problem.

Let us also consider a discrete version of the wormlike chain model, which will allow us to make a connection between the bending rigidity and the statistical segment length. As illustrated in Fig. 2(b), we now represent the chain as a series of bonds i = 1, 2,…, n of length ℓ and corresponding vector r_i. The bending penalty in the discrete model is

$$\frac{E_{\mathrm{b}}}{k_{\mathrm{B}}T}=\sum _{i=1}^{n-1}\kappa (1-\text{cos}\hspace{0.17em}{\theta }_i)$$ (6)

where κ is the bending constant. For two contiguous segments in Fig. 2, the angle between them is computed by

$$\text{cos}\hspace{0.17em}{\theta }_i=\frac{{\mathbf{r}}_i·{\mathbf{r}}_{i+1}}{{\ell }^2}$$ (7)

For clarity, Fig. 2(c) shows how the bending energy in Eq. (6) depends on the angle between two bonds.

The bending constant κ can be converted into a statistical segment length b via [48]

$$\frac{b}{2}=\left(\frac{\kappa }{\kappa -\kappa \coth (\kappa )+1}\right)\ell$$ (8)

The derivation of Eq. (8), which was omitted in our previous work [48], is included here as the Appendix. In the limit of very stiff chains, Eq. (8) reduces to

$$b=2\kappa \ell \hspace{1em}(\kappa \to \infty )$$ (9)

In practice, the latter result is generally a good approximation for $\kappa \mathrm{\gtrsim }10$ because $\coth (\kappa )$ quickly approaches unity.

3. Methods for Measuring the Statistical Segment Length

There are a number of different experimental approaches to measure the statistical segment length or the persistence length of DNA, each with particular advantages and shortcomings. This section reviews a trio of approaches, with the goal of providing an overview (and some key references) for the biomechanics community.

3.1. Light Scattering.

Light scattering is a classical method to measure the radius of gyration, R_g, of a polymer coil [43]. When segment-segment excluded volume is negligible, the radius of gyration is

$$R_{\mathrm{g}}^2=\frac{N{b}^2}{6}$$ (10)

Comparison with Eq. (5) reveals that R_g and R_e differ by a factor of $\sqrt{6}$ and thus are used interchangeably in the polymer literature to denote the size of the chain.

Light scattering involves passing a beam of radiation with wavelength λ through the sample and measuring the scattered radiation at some angle θ, which should not be confused with the bond angle θ_i in Fig. 2. These quantities define the magnitude of the scattering vector in reciprocal space [43]

$$q=\frac{4\pi }{\lambda }\text{sin}\hspace{0.17em}\frac{\theta }{2}$$ (11)

In the limit qR_g ≪ 1, known as the Guinier regime and sketched in Fig. 3, the form factor for the scattered light from an isotropic solution is [43]

Fig. 3.

Schematic illustration of the Guinier regime for light scatting. The radius of the DNA molecule, R_g, is small enough so that it fits well within the wavelength λ of the light. The figure is not drawn to scale; the Guinier regime requires qR_g ≪ 1, where the scattering wave vector magnitude q is defined by Eq. (11).

$$P\left(q\right)=1-\frac{q^2}{3}{R}_{\mathrm{g}}^2+\cdots$$ (12)

While this brief explanation highlights key aspects of light scattering, its implementation in polymer physics is somewhat more involved. The particular method for extracting the radius of gyration from light scattering is known as a Zimm plot; the reader is referred to the monograph by Hiemenz and Lodge [43] for a lucid explanation of this approach.

In a recent paper [2], we compiled a large set of experimental data on light scattering (and neutron scattering) from DNA in solutions with relatively high salt concentrations. When a strong polyelectrolyte like DNA is immersed in a salt solution, counterions in solution condense onto the chain such that the charge spacing is equal to the Bjerrum length, i.e., the Coulombic energy between charges on the chain backbone is equal to thermal energy [49]. This process is known as Manning condensation [50]. If multivalent ions are used, such as spermidine, they can condense the DNA and substantially affect the persistence length [51]. The experiments of interest here use simple salts, such as NaCl, which undergo Manning condensation without compacting the DNA. In the context of the Debye–Hückel formalism, these solutions correspond to high ionic strengths, where electrostatic interactions are screened and the neutral wormlike chain model is expected to be a good approximation. Figure 4 compares simulations of the discrete wormlike chain model to these experimental data [52–63]. We obtained good agreement using a statistical segment length of b = 106 nm.

Fig. 4.

Dependence of the radius of gyration on the number of base pairs of DNA. The x's are experimental light and neutron scattering data [52–63] and the open circles are results from simulations of a discrete wormlike chain model of DNA using b = 106 nm and a hard-core excluded volume interaction with a width w = 4.6 nm [2]. (Reproduced with permission from Tree et al. [2]. Copyright 2013 American Chemical Society).

Light scattering is a very mature experimental technique, and the theory for interpreting the results enjoys widespread acceptance. A particular advantage of light scattering when compared to the methods discussed later is that it is a bulk measurement of DNA and automatically obtains an ensemble average over many chains. The latter aspect conveys important advantages in terms of throughput, while the former ensures that there are no artifacts due to nearby surfaces. One of the challenges in using light scattering for DNA is that the application of Eq. (12) requires that the radius of gyration be small compared to the wavelength of the light, i.e., q⁻¹ ≫ R_g. Since the statistical segment length of DNA obtained from Fig. 4 is already approaching visible wavelengths, it is not possible to investigate the configurations of very large DNA molecules using light scattering. However, it is possible to investigate persistence-length scale features by using radiation wavelengths and scattering angles such that q⁻¹≈ b.

3.2. Optical and Magnetic Tweezers.

The limitation on size that we just noted for light scattering is reversed when using optical or magnetic tweezers. These single-molecule techniques, illustrated schematically in Fig. 5, permit a direct measurement of the force-extension behavior of DNA. In optical tweezer experiments [64], the two ends of the DNA are connected to beads through an antibody-antigen reaction. In the typical setup in Fig. 5(a), a micropipette immobilizes one of the beads while the second bead is moved using a laser, which exerts a radiation pressure on the bead that produces a harmonic potential well. In the standard magnetic tweezer experiment [65] setup illustrated in Fig. 5(b), one end of the DNA is bound to the surface and the distal end is attached to a magnetic bead. The magnetic bead is then pulled from the surface in a magnetic field gradient. (The bead can also be rotated, which makes this setup ideal for examining buckling of DNA under torsion [65].) Since the bead now moves vertical with respect to the focal plane, rather than within the focal plane, its displacement is measured from the diffraction rings.

Fig. 5.

Schematic illustrations of (a) optical tweezers and (b) magnetic tweezers. Neither figure is drawn to scale. The extension X of the DNA molecule is indicated. (a) The DNA is tethered to beads on each end. One bead is held immobile by a micropipette, while the other bead is moved using an optical trap. (b) The DNA is tethered on one end to a coverslip and on the other end to a magnetic bead. A pair of magnets create a magnetic field gradient (dashed lines), which exerts a force on the bead.

The force extension behavior predicted from the wormlike chain model is [41]

$$\frac{F{\ell }_{\mathrm{p}}}{k_{\mathrm{B}}T}=z+\frac{1}{4{(1-z)}^2}-\frac{1}{4}$$ (13)

where z = X/L is the fractional extension of the chain. The latter interpolation formula is often referred to as the “Marko-Siggia” equation and finds widespread use in the modeling of DNA. Much like our discussion of light scattering, Eq. (13) assumes that there are no excluded volume interactions between segments of the DNA. This is a good approximation at strong extensions, but breaks down for weak forces [66].

Figure 6 shows the prediction of Eq. (13) for a 97 kilobase pair (kbp) DNA from a famous tweezer experiment [41,67]. The persistence length obtained from the fit to the Marko–Siggia equation is ℓ_p = 53 nm. Using Eq. (4), we see that this result is in excellent agreement with the statistical segment length obtained by light scattering in Fig. 3.

Fig. 6.

Force-extension of a 97 kbp DNA molecule. The solid line is a fit to Eq. (13) using L = 32.8 μm and ℓ_p = 53 nm, and the dashed line is the prediction of a freely jointed chain model with b = 100 nm. (Reprinted with permission from Marko and Siggia [41]. Copyright 1995 American Chemical Society).

When compared to light scattering, the throughput of an optical tweezer experiment is very low—each DNA molecule must be trapped and then studied one-at-a-time. However, a single molecule can be stretched many times, so outstanding statistics can be obtained at the single molecule level. Moreover, since the magnetic tweezer approach involves surface immobilization and magnetic field gradients can be created over large areas, it is possible to parallelize the magnetic tweezer experiment [68]. A second contrast with light scattering is that the tweezer experiment needs to be designed such that the surfaces do not affect the measurement [69], and the pulling rate needs to be sufficiently slow so that the DNA can equilibrate at each value of the force (or, equivalently, each value of the fractional extension) to avoid nonequilibrium effects.

While we have focused on several limitations of tweezers vis-à-vis light scattering, they have two significant advantages. First, force-extension measurements directly measure the persistence length via Eq. (13), without the need for a somewhat elaborate analysis (using the Zimm plot and several theories) to obtain a statistical segment length from light scattering. Converting from ℓ_p to the statistical segment length b is straightforward by Eq. (4). Second, as already noted in passing at the outset, minimizing the surface interactions typically requires using relatively long DNA molecules. Thus, tweezer measurements are ideally suited to investigate the limit L ≫ b.

3.3. Atomic Force Microscopy (AFM).

In AFM, the DNA are adsorbed onto an atomically flat mica sheet and imaged using a cantilever tip [70,71], illustrated in Fig. 7. The DNA is thus confined to approximately two dimensions, so the analysis of the persistence length changes; the experiment yields a measure of the root-mean-squared end-to-end distance of a two-dimensional polymer. The two-dimensional analog of the Kratky–Porod equation (2) simply requires noting that the persistence length in two dimensions is twice that in three dimensions [70]. As a result, Eq. (2) becomes

Fig. 7.

Schematic illustration of an atomic force microscopy measurement of DNA. The DNA are adsorbed onto a cleaved mica surface and equilibrate in two dimensions. The surface is imaged from the deflections of the cantilever tip. The drawing is not to scale.

$$R_{\mathrm{e}}^2=4{\ell }_{\mathrm{p}}L-8{\ell }_{\mathrm{p}}^2\left[1-\text{exp}\left(-\frac{L}{2{\ell }_{\mathrm{p}}}\right)\right]$$ (14)

In practice, it is easier to measure the decay in the tangent-tangent correlation in two-dimensions [11],

$$⟨\mathbf{t}\left(s\right)·\mathbf{t}(s+\Delta s)⟩=\text{exp}\left(-\frac{\Delta s}{2{\ell }_{\mathrm{p}}}\right)$$ (15)

where t(s) is a unit tangent vector to the chain backbone at position s. The latter quantity is readily measured from the AFM image.

By way of example, Fig. 8 shows AFM images of a 1000 base pair (bp) DNA molecule with different amounts of staining with the intercalating dye YOYO-1. The results of this study led Kundukad et al. to conclude that the persistence length of DNA is ℓ_p = 57 nm, independent of the amount of staining, a point we will revisit shortly.

Fig. 8.

AFM images of DNA (1000 bp) adsorbed onto a cleaved mica surface with different loading of YOYO dye. (Reproduced with permission from Kundukad et al. [11]. Copyright 2014 Royal Society of Chemistry).

Like tweezer experiments, AFM involves interrogating single molecules. As such, the theoretical interpretation of AFM data in terms of Eq. (15) has a similar directness as the force-extension model in Eq. (13), and thus a simpler interpretation than light scattering. Inasmuch as the experiment involves adsorbing the DNA onto the surface, it is important to avoid various artifacts [70–72], most notably kinetic trapping in a nonequilibrium state [70]. AFM measurements are generally limited to small DNA molecules, in order to avoid segment-segment excluded volume interactions that would affect the tangent-tangent correlation. A possible limitation in these experiments is the sequence-dependence of the DNA bending rigidity [73,74], as well as particular sequences with intrinsic curvature [75]. It is important to control for sequence-dependent effects that can affect measurements of the backbone tangent correlations at short length scales.

4. Effect of Bisintercalation by YOYO

Kundukad et al. [11] recently provided a concise review of the relevant literature for the effect of YOYO-1 on the DNA statistical segment length in the context of presenting their new experimental data. There is consensus that the dye-complexed DNA has a longer contour length L than native DNA. (This may seem to contradict the standard dogma for synthetic polymers, where the contour length is computed from the carbon-carbon bonds along the backbone. For DNA, the contour length is generally defined in terms of the rise between base pairs, which can increase when the intercalating molecule is inserted between bases.) However, the effect of the dye on the statistical segment length of DNA remains an open question and seems to depend on the method of measurement. The following two paragraphs summarize the review of tweezer-based and AFM measurements by Kundukad et al. [11].

When the DNA is stretched using optical tweezers or magnetic tweezers, a large number of early studies [4–7] indicated a substantial decrease in the statistical segment length due to intercalation, up to around 70%. One of the key challenges in stretching experiments is maintaining the DNA-YOYO complex at equilibrium, and large hysteresis effects often occur. The force-extension behavior of the DNA, embodied in Eq. (13), is single-valued. As a result, if the same force gives rise to two different values of the extension, then the experiment is not probing the equilibrium behavior of the polymer. A more recent magnetic tweezers study by Günther et al. [8] claimed to maintain the dye-complexed DNA at equilibrium by using a moderate ionic strength to promote stable and stoichiometric binding of YOYO-1 to the DNA. They further used magnetic tweezers at relatively low forces (0.1–10 pN) to avoid reaching the higher forces where YOYO-1 binding becomes force-dependent [8]. These experiments [8] indicated no change in statistical segment length due to dye loading with a statistical segment length b around 104 nm. In all of these cases [4–8], the analysis assumes that the force-extension curve for DNA can be fit using Eq. (13), with the contour length and persistence length being fitting parameters.

As noted in Sec. 3.3, an alternate approach to measure the statistical segment length is to adsorb the DNA onto a surface and use atomic force microscopy (AFM) to determine the persistence length (and thus the statistical segment length) by using either the mean-square end-to-end distance or the tangent auto-correlation function. AFM measurements are also in disagreement, with an early experiment [10] indicating a 46% decrease in statistical segment length due to dye loading and the most recent experiment [11] indicating a constant statistical segment length around b = 114 nm at the dye saturation level of one molecule per four basepairs of DNA. The advantage of AFM, relative to the tweezer experiments, is that the DNA is not under tension. As noted in the preceding discussion of this method in Sec. 3.3, AFM measurements assume that the presence of the surface only confines the chain to two dimensions without introducing any other perturbation to the chain, either equilibrium (due to inhomogeneous surface properties) or nonequilibrium (kinetically trapping the chain during the adsorption process). Indeed, Kundukad et al. [11] point out that previous AFM results by Maaloum et al. [10] were likely affected by excluded volume effects that are not included in Eq. (14).

Taken together, the magnetic tweezer results by Günther et al. [8] and the AFM results from Kundukad et al. [11] strongly suggest that the persistence length (and thus the statistical segment length) of DNA is not affected by intercalation by YOYO. While we are not aware of any groups that have reproduced (and published) either of these studies, their identification and systematic elimination of various artifacts in the previous work provides confidence in the robustness of their conclusions.

Our own experiments on DNA confinement in nanochannels further support the conclusion that the DNA statistical segment length is unaffected by intercalation by YOYO. When confined, the DNA will stretch along the channel axis due to excluded volume interactions if the channel size D is larger than the statistical segment length [76]. In the opposite situation, where the chain is strongly confined, the DNA again stretches down the channel axis but now due to the bending energy penalty for forming a hairpin [77]. The region D ≈ ℓ_p is particularly interesting because the fractional extension of the chain exhibits a very strong dependence on the ratio D/ℓ_p [29,30,32], and this is the technologically relevant channel size for genomic mapping [22].

In a pair of comparisons [38,39] between experimental data on DNA stretching by confinement in a nanochannel and predictions of a discrete wormlike chain model [78] in the same confinement, we found that this model could predict the experimental data to within about 10% error using the statistical segment length obtained from Fig. 6 [67] and Dobrynin's correction for ionic strength [79]. Related work by Iarko et al. supports this conclusion [40]. Possible sources of the remaining error between theories and experiments include uncertainties in the DNA-wall interactions [30,80,81], anisotropic channel sizes [29,35,82], and challenges in having DNA satisfy the various inequalities governing the underlying scaling theories [82].

5. Perspectives for Future Research

Our current understanding of the literature suggests that YOYO intercalation does not change the statistical segment length of DNA, supported by well-controlled AFM [11] and magnetic tweezer experiments [8], as well as the relatively good agreement between simulations and experiments for DNA extension in nanochannels [38–40]. That intercalation leaves the bending energy unaffected is a remarkable result, especially in light of the general consensus that DNA intercalation leads to an increase in the rise of DNA, and thus an increase in the contour length L appearing in the wormlike chain model [11]. In particular, if the bending energy of DNA comes from stacking interactions between the bases, disrupting the stacking should make it easier to bend the molecule.

Going forward, there are opportunities to advance on both the experimental end and via biomechanical modeling. On the experimental side, we are not aware of any attempts to use scattering to investigate the role of YOYO intercalation, although there is one report using dynamic light scattering to study the interactions of DNA with diaminobenzidine, a partial intercalator [83]. Note that such an experiment is very difficult to interpret using the Guinier regime illustrated in Fig. 3. While these experiments would furnish a value of the radius of gyration, it would be challenging to determine whether the radius of gyration changed due to an increase in contour length L or a decrease in statistical segment length b. An alternate approach is to move to smaller wavelength radiation, such as X-rays, that directly probes chain statistics at the length scale of the statistical segment length, and check whether the scattering pattern changes with YOYO loading. While these experiments may be technically demanding, it would be very satisfying to confirm that YOYO labeling does not affect the DNA statistical segment length with a classic bulk measurement method.

In addition to the verifying the single-molecule measurements via scattering, it still remains to establish the underlying physics for the increase in L and invariance in b upon intercalation. For example, is the increased contour length at fixed statistical segment length the result of unwinding of the DNA by the intercalation [11]? This is inherently a multiscale problem and an area where biomechanical modeling could play a key role. The modeling problem is quite challenging—one needs to first understand the local chemical interactions between the dyes and the DNA, and then coarse-grain those interactions into a mesoscale model. One possible avenue at the coarse-grained scale is to use a three-sites-per-nucleotide (3SPN) model [84,85], and reparameterize it from atomistic simulations in a manner similar to the way this coarse-grained model was modified to account for solvation [86] and ions [87]. The coarse-grained model can then be simulated for reasonable molecular weights, of order 100 bp, to obtain the backbone tangent correlations in the three-dimensional analog to Eq. (15). While such simulations will be computationally intense, they are a very promising avenue for unraveling the mechanistic basis for the role of bisintercalation on the statistical segment length of DNA.

Appendix Derivation of the Relationship between the Bending Constant and Persistence Length

Following Heimenz and Lodge [43], the persistence length can be computed from

$${\ell }_{\mathrm{p}}=(C_{\infty }+1)\frac{\ell }{2}$$ (A1)

where

$$C_{\infty }=\frac{1+⟨\hspace{0.17em}\text{cos}\hspace{0.17em}\theta ⟩}{1-⟨\hspace{0.17em}\text{cos}\hspace{0.17em}\theta ⟩}$$ (A2)

is known as the limiting value of the characteristic ratio. From Eq. (6), the average angle is governed by Boltzmann statistics

$$⟨\hspace{0.17em}\text{cos}\hspace{0.17em}\theta ⟩=\frac{{\int }_0^{\pi }\hspace{0.17em}\text{cos}\hspace{0.17em}\theta \hspace{0.17em}\text{exp}[-\kappa (1-\text{cos}\hspace{0.17em}\theta \left)\right]\text{sin}\hspace{0.17em}\theta \hspace{0.17em}d\theta }{{\int }_0^{\pi }\hspace{0.17em}\text{exp}\hspace{0.17em}[-\kappa (1-\text{cos}\hspace{0.17em}\theta \left)\right]\text{sin}\hspace{0.17em}\theta \hspace{0.17em}d\theta }$$ (A3)

These integrals can be evaluated analytically, leading to

$$⟨\hspace{0.17em}\text{cos}\hspace{0.17em}\theta ⟩=\coth \kappa -{\kappa }^{-1}$$ (A4)

Using Eq. (A4) in Eq. (A2) and inserting the result into Eq. (A1) leads to Eq. (8).

In the previous work [2,78], we have used the approximation ℓ_p ≈ C_∞ℓ/2 in place of the more accurate Eq. (A1). Repeating the derivation leads to

$${\ell }_{\mathrm{p}}=\left(\frac{\kappa -1+\kappa \coth \kappa }{\kappa +1-\kappa \coth \kappa }\right)\frac{\ell }{2}$$ (A5)

which is also a reasonable approximation for relatively stiff chains.

Nomenclature

b =

statistical segment length

bp =

base pairs

C_∞ =

limiting value of the characteristic ratio

D =

channel size (diameter)

g =

global persistence length

k_B =

Boltzmann's constant

kbp =

kilobase pairs

ℓ =

bond length

ℓ_p =

persistence length

L =

contour length

Mb =

megabase pairs

n =

number of bonds

N =

number of statistical segment lengths

P(q) =

form factor

q =

magnitude of the scattering wave vector

r(s) =

location of monomer s

R_e =

end-to-end distance

R_g =

radius of gyration

s =

location along the chain contour

t =

tangent vector

T =

temperature

w =

effective width

X =

extension

z =

fractional extension

θ =

scattering angle

θ_i =

bond angle

κ =

bending constant

λ =

wavelength