Fundamental length scales

The smallest of the length scales characterizing the conformations of DNA is the width of the DNA backbone. While the steric dimension of the backbone is approximately 2 nm, electrostatic repulsion arising from the charged phosphate groups makes the DNA backbone appear thicker than would be expected, were the DNA a neutral polymer. The standard approach to treat the electrostatics is to consider an effective width w (Stigter, Reference Stigter1975, Reference Stigter1977), which is obtained from a mapping of the osmotic pressure produced by a suspension of charged cylinders of diameter d to that produced by neutral cylinders of ‘effective width’ w. For the high ionic strength buffers sometimes used in experiments of confined DNA, for example 100 mM of monovalent salt, Stigter's theory (Stigter, Reference Stigter1977) produces an effective width of approximately 5 nm. At lower ionic strength, the effective width is available from Stigter's theory and numerical results have been reported, for example, by Hsieh et al. (Reference Hsieh, Balducci and Doyle2008) or Tree et al. (Reference Tree, Muralidhar, Doyle and Dorfman2013a). When making this mapping, the details of the electrostatic interactions are replaced by an equivalent neutral polymer model.

At the next level up is the DNA persistence length l_p, which reflects the decay of the correlations between the tangent vectors along the DNA backbone. A qualitative (but useful) way to think about the persistence length is that it characterizes the ability of the DNA to be bent by thermal energy. For high ionic strength buffers, the persistence length of DNA is approximately 50 nm, a frequently cited result obtained by Bustamante et al. using magnetic tweezers (Bustamante et al., Reference Bustamante, Marko, Siggia and Smith1994). The persistence length is also affected by ionic strength, since the ions in solution screen the charges along the DNA backbone and thus affect its propensity to bend under thermal energy (Odijk, Reference Odijk1977; Skolnick and Fixman, Reference Skolnick and Fixman1977). Similar to the effective width, it is convenient to subsume the details of the electrostatic interactions into a persistence length that captures both the intrinsic contributions, arising from the steric interactions, and an electrostatic contribution. After the mapping is complete, we now have an equivalent neutral polymer model for the DNA that is thicker (due to the effective width) and stiffer (due to the electrostatic contribution to the persistence length) than would be expected based solely on the steric volume of its constituent atoms.

To account for electrostatic contributions to the persistence length, Dobrynin (Reference Dobrynin2005) proposed a semi-empirical formula to correct for the ionic strength dependence, correcting an approximation in the classic Odijk–Skolnick–Fixman (OSF) theory by Odijk (Reference Odijk1977) and Skolnick and Fixman (Reference Skolnick and Fixman1977). Hsieh et al. (Reference Hsieh, Balducci and Doyle2008) showed that the Dobrynin model provides a superior description of DNA confined to a nanoslit when compared to the OSF theory. Subsequently, Chuang et al. (Reference Chuang, Reifenberger, Cao and Dorfman2017) extended Dobrynin's result to further incorporate sequence effects. Their analysis of the stretching of human DNA in 45 nm nanochannels, when combined with Odijk's theory for strong confinement (Odijk, Reference Odijk1983; Burkhardt et al., Reference Burkhardt, Yang and Gompper2010) and Dobrynin's theory (Dobrynin, Reference Dobrynin2005), yielded the useful formula,

(1)$$l_p\;[ {{\rm nm}} ] = ( {23 + 33\gamma + 26\gamma^2} ) + \displaystyle{{1.9195} \over {\sqrt {I[ {\rm M} ] } }}, \;$$

where γ is the fraction of G-C base pairs. In the second term, which is the electrostatic contribution attributed to Dobrynin (Reference Dobrynin2005), I is the ionic strength of the buffer. The details surrounding the role of electrostatics and counterion chemistry on the persistence length remain a subject of investigation (Brunet et al., Reference Brunet, Tardin, Salomé, Rousseau, Destainville and Manghi2015; Trizac and Shen, Reference Trizac and Shen2016; Guilbaud et al., Reference Guilbaud, Salome, Destainville, Manghi and Tardin2019), and their impact on nanochannel DNA confinement is also complicated (Chuang et al., Reference Chuang, Reifenberger, Bhandari and Dorfman2019). These additional factors affect the quantitative agreement between theory and experiment.

At the longest length scale, at least for problems of interest to us here, is the DNA contour length L. At first glance, this appears to be a very straightforward quantity to compute; simply multiply the number of base pairs by the 0.34 nm/bp rise of B-DNA. However, fluorescence microscopy experiments typically require labeling of the DNA backbone with an intercalating dye, such as YOYO-1, that increases the extension of the DNA (Perkins et al., Reference Perkins, Smith, Larson and Chu1995; Bakajin et al., Reference Bakajin, Duke, Chou, Chan, Austin and Cox1998; Randall and Doyle, Reference Randall and Doyle2005; Jeffet et al., Reference Jeffet, Kobo, Su, Grunwald, Green, Nilsson, Eisenberg, Ambjörnsson, Westerlund, Weinhold, Shabat, Purohit and Ebenstein2016; Lee et al., Reference Lee, Lee, Kim, Wang, Park, Jung, Chen, Chang, Ikeda, Sugiyama and Jo2018b). At full intercalation, the DNA contour length can increase by circa 20–30% (Dorfman, Reference Dorfman2010; Nyberg et al., Reference Nyberg, Persson, Akerman and Westerlund2013). Kundukad et al. (Reference Kundukad, Yan and Doyle2014) provide a concise review of different reports for DNA contour length increase upon YOYO-1 intercalation; we have found it convenient to assume that each bound dye molecule adds 0.51 nm to the DNA rise (Gupta et al., Reference Gupta, Bhandari and Dorfman2018).

Derived length scales

While the trio of the contour length, persistence length, and effective width describe the length scales of the DNA chain, particular combinations of these parameters are often more useful for describing the configurations of a polymer. For a particularly lucid discussion of polymer conformations in general, consult the Perspectives article by Wang (Reference Wang2017). For the moment, we will focus on the relevant parameters for understanding DNA confinement, adopting typical length scales w = 5 nm and l_p = 50 nm for model calculations. We also further focus on the limit of chains where L  ≫ l _p, which are coils rather than rods (Hiemenz and Lodge, Reference Hiemenz and Lodge2007). For λ-DNA (48.5 kbp), which is a commonly used model system, the stained contour length at full intercalation is close to 20 μm (Tegenfeldt et al., Reference Tegenfeldt, Prinz, Cao, Chou, Reisner, Riehn, Wang, Cox, Sturm, Silberzan and Austin2004), corresponding to 400 persistence lengths assuming no effect of the dye on the persistence length (Gunther et al., Reference Gunther, Mertig and Seidel2010; Kundukad et al., Reference Kundukad, Yan and Doyle2014; Dorfman, Reference Dorfman2018).

The size of the coiled chain is characterized by either the end-to-end distance or the radius of gyration, R_g, of the polymer. While either metric is suitable for our purposes, we will focus here on the radius of gyration. For an ideal polymer, the radius of gyration is given by random walk statistics,

(2)$$R_g^2 = \displaystyle{{Nb^2} \over 6}, \;$$

where N is the number of statistical segments and b is the statistical segment length (Hiemenz and Lodge, Reference Hiemenz and Lodge2007); it follows that L = Nb. The exponent characterizing the scaling for the radius of gyration of an ideal chain is thus N ^1/2. For a semiflexible chain like DNA, the correspondence between the random walk statistics and the Kratky and Porod (Reference Kratky and Porod1949) result for the end-to-end distance of a long wormlike chain gives b = 2l _p.

When excluded volume is important, the chain obeys self-avoiding random walk statistics with a radius of gyration

(3)$$R_g\cong L^{3/5}l_p^{1/5} w^{1/5}.$$

Comparing Eqs. (2) and (3), the exponent for the scaling of the polymer size changes from N ^1/2 for an ideal chain to $N^\nu$ in the presence of excluded volume, where we have made the trivial conversion between contour length and degree of polymerization, L ~ N. The parameter ν is known as the Flory exponent, and accounts for the additional swelling of the polymer coil caused by the excluded volume between distal segments. In Eq. (3), we have used the classical value of the Flory exponent ν = 3/5 to quantify the scaling for the chain size; the most accurate estimate of the scaling exponent by Clisby is ν = 0.587597(7) (Clisby, Reference Clisby2010).

When is excluded volume important? To answer this question, we first need to understand the excluded volume interactions between different segments of the DNA. The cross-over is typically written in terms of an excluded volume parameter (Wang, Reference Wang2017),

(4)$$z = \left({\displaystyle{3 \over {2\pi }}} \right)^{3/2}\bigg({\displaystyle {{v_{ex}} \over {b^3}}} \bigg)N^{1/2}.$$

The Onsager excluded volume, v_ex, between two cylindrical rods of width w and length l_p is approximately $v_{ex}\cong l_p^2 w$. To determine the length scale over which excluded volume matters, we set z ≅ 1. The number of statistical segments required to satisfy this condition is then $b^6/v_{ex}^2$. Using Eq. (2), the thermal blob size,

(5)$$l_b\cong \displaystyle{{b^4} \over {v_{ex}}}$$

is defined as the size of a polymer chain where z ≅ 1, which is conventionally taken as the point at which excluded volume becomes important. In the standard polymer physics interpretation (Wang, Reference Wang2017), a polymer whose radius of gyration exceeds the thermal blob size is affected by excluded volume.

While these scaling arguments are useful for establishing the general phenomena, the neglect of various prefactors makes them less useful for modeling DNA in nanochannel experiments. Indeed, Tree et al. (Reference Tree, Muralidhar, Doyle and Dorfman2013a) pointed out that there are several equally plausible definitions of the thermal blob size and these definitions give remarkably different results for the onset of excluded volume for DNA, ranging from 16.8 to 166 kbp. These values span the range of typical DNA sizes used in experiments. To be more quantitative, Tree et al. (Reference Tree, Muralidhar, Doyle and Dorfman2013a) instead computed the Flory exponent for DNA as a function of the number of base pairs in the relationship R _g ~ N ^υ, leveraging the renormalization group result from Chen and Noolandi (Reference Chen and Noolandi1992). Figure 2 shows that typical molecular size standards, such as λ-DNA, obey neither Eq. (2) nor Eq. (3). Rather, typical DNA sizes used in the experiments discussed here are in a very broad cross-over between ideal chain scaling and self-avoiding random walks.

Fig. 2. Flory exponent ν as a function of the number of statistical segments obtained from renormalization group calculations (Chen and Noolandi, Reference Chen and Noolandi1992). The curves correspond to different values of w/b, where the upper (red) dashed line is a typical value for single-stranded DNA and the lower (blue) dashed line is a typical value for double-stranded DNA. The locations for λ-DNA (48.5 kb) and T4-DNA (166 kb) are denoted by boxes. The asymptotic limits of a rod, random walk (Gaussian chain), and self-avoiding random walk (swollen chain) are indicated. Reproduced with permission from (Tree et al., Reference Tree, Muralidhar, Doyle and Dorfman2013a), © 2013 American Chemical Society.

In what follows, we will define the thermal blob size,

(6)$$l_b\cong \displaystyle{{l_p^2 } \over w}$$

as this definition has proven to be very useful for defining the onset of the de Gennes regime for a confined polymer. For DNA in a high ionic strength buffer, this definition of the thermal blob size is approximately 500 nm, corresponding to approximately 2.5 μm of DNA. This length scale will play an important role in defining the extent of confinement in what follows.

Wall depletion length

When DNA is confined to a nanochannel, there is an additional length scale characterizing the DNA–wall interactions. Both the DNA and the walls are negatively charged and therefore repel one another. In the simplest model, one can capture this effect by introducing a depletion length δ such that the effective channel size is D_eff = D−δ. Several models have been proposed for the depletion length, including using the DNA effective width w (Wang et al., Reference Wang, Tree and Dorfman2011), a rigid-rod approximation (Reisner et al., Reference Reisner, Pedersen and Austin2012), and a semiflexible chain model (Reisner et al., Reference Reisner, Pedersen and Austin2012). Bhandari et al. (Reference Bhandari, Reifenberger, Chuang, Cao and Dorfman2018) measured this depletion length at different ionic strengths using a high-throughput genome mapping approach and found that their data were described by the simple phenomenological model δ = 6.5 λ_D + 0.64, where λ_D is the Debye length of the buffer solution.

De Gennes regime (weak confinement)

Daoud and De Gennes (Reference Daoud and De Gennes1977) developed the classic treatment of the extension of weakly confined polymers. By weak confinement, we mean that the channel size is smaller than the radius of gyration (so that the DNA needs to extend along the channel axis), but larger than the thermal blob size in Eq. (6). The latter restriction implies that the subchain within a correlation length D is itself a self-avoiding walk that obeys Eq. (2). The contour length of the subchain, which is also referred to as a blob and indicated by the circle in Fig. 3, is given by $L_{{\rm blob}}\cong D^{5/3}l_p^{{-}1/3} w^{{-}1/3}.$ The number of blobs is N _blob = L/L _blob and the mean span of the chain is X ≅ N _blobD. Putting these results together gives the de Gennes scaling for a confined polymer,

(9)$$X\cong L\left({\displaystyle{{wl_p} \over {D^2}}} \right)^{1/3}.$$

The variance in the chain size is σ ² ≅ N _blobD ², which gives (Dai and Doyle, Reference Dai and Doyle2013)

(10)$$\sigma ^2\cong L( {Dwl_p} ) ^{1/3}.$$

While these results are derived in the context of blob theory, equivalent results are obtained from a modified Flory theory (Jun et al., Reference Jun, Thirumalai and Ha2008; Wang et al., Reference Wang, Tree and Dorfman2011). The prefactors for the scaling laws can be computed via simulations (see, e.g. Muralidhar et al., Reference Muralidhar, Tree, Wang and Dorfman2014b), but they are expected to be O(1) quantities. For the purpose of understanding experimental data, the exact value of the prefactor is not particularly important.

Odijk regime (strong confinement)

Odijk (Reference Odijk1983) considered the opposite case to Daoud and de Gennes, where the polymer is strongly confined in a channel that is much smaller than its persistence length. Here, rather than forming a series of blobs, Odijk showed that the DNA instead forms a chain of deflection segments of characteristic size $\lambda \cong D^{2/3}l_p^{1/3}$, which is known as the deflection length. In essence, the chain performs a persistent random walk due to its stiffness, but the collisions with the hard wall over distances corresponding to the deflection length lead to decorrelations between different segments of the chain.

In the absence of any hairpin turns, the extension of this chain of deflection segments is

(11)$$\left\langle X \right\rangle = L\left[{1-0.18274{\left({\displaystyle{D \over {l_p}}} \right)}^{2/3}} \right], \;$$

where the prefactor for a square channel was computed by Burkhardt et al. (Reference Burkhardt, Yang and Gompper2010) from an accelerated particle model. These authors further computed the variance in the chain extension, arriving at

(12)$$\sigma ^2 = 0.00956\displaystyle{{D^2} \over {l_p}}L.$$

Corrections for rectangular channels (Burkhardt et al., Reference Burkhardt, Yang and Gompper2010) and circular channels (Yang et al., Reference Yang, Burkhardt and Gompper2007) are available, and slightly different results have been obtained from different calculations (Chen, Reference Chen2016).

The statistics for strongly confined chains in the Odijk regime and weakly confined chains in the de Gennes regime are markedly different. In the de Gennes regime, the chain is only weakly perturbed from its coil shape and the dependence of the chain extension on channel size is strong. In contrast, the Odijk regime corresponds to almost completely extended chains, since the energetic cost to form a hairpin is extremely high (Odijk, Reference Odijk2006). The extension of the chain in the Odijk regime thus has a very weak dependence on the channel size. The increase in extension as the channel size decreases in the Odijk regime is driven by alignment of the deflection segments along the channel axis, which increases as the channel size decreases. This orientation effect is also the reason why the variance in the Odijk regime decreases with decreasing channel size; as the alignment fluctuations are suppressed, the fluctuations in the chain extension are also suppressed. The fluctuations in the chain extension also decrease with decreasing channel size in the de Gennes regime, albeit more weakly. The reason for the weaker decay in fluctuations is that these fluctuations are driven by the blobs, not by orientation of the chain with the channel axis.

Additional scaling regimes at intermediate confinement

While the de Gennes and Odijk regimes possess an aesthetic beauty as physical models, they are of little use in describing the extension of DNA in confinement. If we take a thermal blob size of 500 nm, then a conservative estimate of the start of the de Gennes regime is a channel that is several microns in size. However, in order to reach de Gennes statistics, the DNA in such a large channel needs to comprise several blobs (Muralidhar et al., Reference Muralidhar, Tree, Wang and Dorfman2014b). If we use an aggressive set of bounds and set D = 1 μm and only one blob, we get a minimum contour length of L = 16 μm. In principle, we would like to be at least an order of magnitude away from this boundary, which would imply we need circa 500 kbp long DNA. While such long molecules have been used in some physics experiments (Tegenfeldt et al., Reference Tegenfeldt, Prinz, Cao, Chou, Reisner, Riehn, Wang, Cox, Sturm, Silberzan and Austin2004; Freitag et al., Reference Freitag, Noble, Fritzsche, Persson, Reiter-Schad, Nilsson, Graneli, Ambjornsson, Mir and Tegenfeldt2015) and are routinely used for genome mapping in nanochannels (Müller and Westerlund, Reference Müller and Westerlund2017), most physical experiments use shorter DNA and/or narrower channels.

At the other end of the spectrum, the Odijk regime requires that the channel size be much smaller than the persistence length. For high ionic strengths, the requisite channel sizes are below the 20 nm lower bound for injecting DNA (Cao et al., Reference Cao, Yu, Wang, Tegenfeldt, Austin, Chen, Wu and Chou2002b) and the smallest channels used for measuring DNA extension (Reisner et al., Reference Reisner, Morton, Riehn, Wang, Yu, Rosen, Sturm, Chou, Frey and Austin2005). It is possible to work with wider channels and remain in the Odijk regime by decreasing the ionic strength (Jo et al., Reference Jo, Dhingra, Odijk, De Pablo, Graham, Runnheim, Forrest and Schwartz2007; Kim et al., Reference Kim, Kim, Kounovsky, Chang, Jung, Depablo, Jo and Schwartz2011; Kounovsky-Shafer et al., Reference Kounovsky-Shafer, Hernandez-Ortiz, Potamousis, Tsvid, Place, Ravindran, Jo, Zhou, Odijk, De Pablo and Schwartz2017; Lee et al., Reference Lee, Lee, Kim, Wang, Park, Jung, Chen, Chang, Ikeda, Sugiyama and Jo2018b), and Odijk's theory seems to be a good description there. However, one does need to be careful in interpreting the data because the electrostatic double-layers from the walls now occupy a substantial fraction of the channel cross-section, complicating the mapping from the polyelectrolyte problem to the neutral polymer model embodied by Odijk's theory (Reisner et al., Reference Reisner, Pedersen and Austin2012; Cheong et al., Reference Cheong, Li and Dorfman2017; Chuang et al., Reference Chuang, Reifenberger, Bhandari and Dorfman2019; Bhandari and Dorfman, Reference Bhandari and Dorfman2020). From a fundamental perspective, the best way to probe the Odijk regime using biopolymers is to switch to stiffer molecules, such as actin (Noding and Koster, Reference Noding and Koster2012) or RecA filaments (Frykholm et al., Reference Frykholm, Alizadehheidari, Fritzsche, Wigenius, Modesti, Persson and Westerlund2014).

Using scaling theory, Odijk (Reference Odijk2008) identified two additional regimes that should exist between what are now referred to as the classic de Gennes regime and the classic Odijk regime. As the channel size decreases below the thermal blob size (~500 nm) but remains above the statistical segment length (~100 nm), the blobs become ellipsoidal, as indicated in Fig. 3, and their statistics are marginal, i.e. the excluded volume corresponds to z ≅ 1. This regime of confinement is often referred to as the ‘extended de Gennes’ regime (Wang et al., Reference Wang, Tree and Dorfman2011) because the scaling for the extension given by Eq. (9) ‘extends’ to even smaller channel sizes. However, the variance of the chain extension is markedly different in the extended de Gennes regime,

(13)$$\sigma ^2 = Ll_p.$$

The latter result, which can be derived using either blob theory (Dai and Doyle, Reference Dai and Doyle2013) or modified Flory theory (Wang et al., Reference Wang, Tree and Dorfman2011), is remarkable – the fluctuations of the confined, real chain are equal to those of an unconfined, ideal chain of the same contour length. The independence of the variance from the channel size was a key to providing affirmative proof for the existence of the extended de Gennes regime in simulations (Dai and Doyle, Reference Dai and Doyle2013) and experiments (Gupta et al., Reference Gupta, Miller, Muralidhar, Mahshid, Reisner and Dorfman2015; Iarko et al., Reference Iarko, Werner, Nyberg, Muller, Fritzsche, Ambjornsson, Beech, Tegenfeldt, Mehlig, Westerlund and Mehlig2015). However, the best theoretical evidence in support of the extended de Gennes regime lies in the excess free energy of confinement (Dai et al., Reference Dai, van der Maarel and Doyle2014), which put to rest a lingering discussion over the existence (or lack thereof) of this regime (Tree et al., Reference Tree, Wang and Dorfman2013b), but this small change in free energy is not straightforward to measure via experiments.

The extended de Gennes regime has a strong connection with the theory of marginal solutions of semiflexible polyelectrolytes (Schaefer et al., Reference Schaefer, Joanny and Pincus1980; Dai et al., Reference Dai, van der Maarel and Doyle2014; Wang, Reference Wang2017). Werner and Mehlig (Reference Werner and Mehlig2014) also recognized a connection between the statistics of the extended de Gennes regime and weakly self-avoiding random walks. Leveraging exact results for weakly self-avoiding walks in the mathematical literature, they were able to provide exact prefactors for both the extension and variance in the extended de Gennes regime. Data from subsequent experiments (Gupta et al., Reference Gupta, Miller, Muralidhar, Mahshid, Reisner and Dorfman2015; Iarko et al., Reference Iarko, Werner, Nyberg, Muller, Fritzsche, Ambjornsson, Beech, Tegenfeldt, Mehlig, Westerlund and Mehlig2015) were in good but not quantitative agreement with the theoretical predictions. Subsequently, Schotzinger et al. (Reference Schotzinger, Menard and Ramsey2020) provided conclusive evidence for the extension scaling in the extended de Gennes regime, explaining previous disagreements between theory and experiment as the result of inadequate characterization of the channel cross-section that were circumvented in their experiments by using a focused-ion beam approach to channel fabrication (Menard and Ramsey, Reference Menard and Ramsey2013).

When the channel size is below the statistical segment length (~100 nm), Odijk (Reference Odijk2008) predicted a cross-over from blob statistics to deflection segments. However, unless the channel size is much smaller than the persistence length, the chain can still form the hairpins indicated in Fig. 3 because the energy of formation is not particularly high (Polson, Reference Polson2018), a phenomenon identified via simulations by Cifra and Bleha (Reference Cifra and Bleha2012) and Dai et al. (Reference Dai, Ng, Doyle and van der Maarel2012). Odijk (Reference Odijk2006) called the length scale for forming a hairpin the global persistence length g.

The presence of hairpins within the chain invalidates the results in Eqs. (11) and (12). When the chain can form hairpins, Odijk showed that its statistics are described by a different scaling variable (Odijk, Reference Odijk2008; Muralidhar et al., Reference Muralidhar, Tree and Dorfman2014a)

(14)$$\xi = \displaystyle{{gw} \over {D^{5/3}l_p^{1/3} }}$$

that measures the total excluded volume created by a hairpin to the spatial volume gD ² occupied by that hairpin. Using a Flory argument, Odijk (Reference Odijk2008) then derived the scaling

(15)$$\left\langle X \right\rangle \cong L\xi ^{1/3}$$

for the chain extension when L ≫ g, i.e. when the chain can form many hairpin folds. The concomitant variance in the chain extension is

(16)$$\sigma ^2\cong gL.$$

Comparing Eqs. (16) and (13) shows that the variance in the extended de Gennes regime and backfolded Odijk regimes is of the same form, but the persistence length of the polymer needs to be replaced by the global persistence length to describe the backfolded Odijk regime.

Muralidhar et al. (Muralidhar et al., Reference Muralidhar, Tree and Dorfman2014a, Reference Muralidhar, Quevillon and Dorfman2016; Muralidhar and Dorfman, Reference Muralidhar and Dorfman2016) undertook systematic studies of the backfolded Odijk regime via simulations. In addition to providing a pedagogical introduction (Muralidhar et al., Reference Muralidhar, Tree and Dorfman2014a) to Odijk's scaling theory (Odijk, Reference Odijk2008), they confirmed all predictions of Odijk's theory for square, rectangular, and circular channels. Importantly, the simulation work by Muralidhar et al. (Reference Muralidhar, Tree and Dorfman2014a) identified a shortcoming in Odijk's theory for the global persistence length (Odijk, Reference Odijk2006), which plays a key role in the overall theory. The disagreement between Odijk (Reference Odijk2006) and Muralidhar et al. (Reference Muralidhar, Tree and Dorfman2014a) was resolved by Chen (Reference Chen2017), who provided a complete correction to Odijk's theory.

Owing to the very large increase in the global persistence length as the channel size decreases (Odijk, Reference Odijk2006), Eq. (16) predicts an equally large increase in the fluctuations of the extension in this regime. These large fluctuations were not realized in experiments with DNA because the DNA was too short to form the large hairpins predicted by Odijk's theory (Su et al., Reference Su, Das, Xiao and Purohit2011). Indeed, experiments to probe the backfolded Odijk regime using DNA are infeasible (Muralidhar et al., Reference Muralidhar, Tree and Dorfman2014a).

Telegraph model for intermediate confinement

The schematic illustration of different regimes in Fig. 3 is not entirely correct; the boundaries are listed as inequalities but, strictly speaking, correspond to strong inequalities. The relatively good agreement between theory and experiment for the extended de Gennes regime (Gupta et al., Reference Gupta, Miller, Muralidhar, Mahshid, Reisner and Dorfman2015; Iarko et al., Reference Iarko, Werner, Nyberg, Muller, Fritzsche, Ambjornsson, Beech, Tegenfeldt, Mehlig, Westerlund and Mehlig2015; Schotzinger et al., Reference Schotzinger, Menard and Ramsey2020) suggests that those inequalities are not especially strong. However, there are no experimental data to support the existence of a backfolded Odijk regime for DNA. While it is possible to probe the Odijk excluded volume by a non-equilibrium measurement (Krog et al., Reference Krog, Alizadehheidari, Werner, Bikkarolla, Tegenfeldt, Mehlig, Lomholt, Westerlund and Ambjornsson2018; Reifenberger et al., Reference Reifenberger, Cao and Dorfman2018), we seem to be again stuck in a situation where there is an elegant scaling theory that says little about the problem of DNA confinement.

Werner et al. (Reference Werner, Cheong, Gupta, Dorfman and Mehlig2017) recognized that the exact theory for the extended de Gennes regime (Werner and Mehlig, Reference Werner and Mehlig2014) applies for all confinements below the thermal blob size but outside the Odijk regime. In other words, there is no reason to distinguish between the deflection segments in the backfolded Odijk regime and the blobs in the extended de Gennes regime. Rather, the key to understanding intermediate confinement is to recognize the weakness of excluded volume interactions for$\;D\ll l_p^2 /w$. While the chain can fold back on itself in these smaller channels, the long distance along the contour length causes the chain to ‘forget’ the previous configurations. As a result, the excluded volume interactions can be treated in a mean-field sense (Chen, Reference Chen2018).

Thus, instead of relying on the extended de Gennes or backfolded Odijk models, the DNA statistics for $D\ll l_p^2 /w$ can be described by the weakly-correlated telegraph model. The telegraph model describes a random walk that moves at a fixed velocity with the random changes in direction indicated in Fig. 4. In the weakly correlated telegraph model, there is a small energy penalty $\varepsilon$ when the chain revisits a lattice site. This one-dimensional model is easily simulated to produce predictions for X and σ ². Importantly, the telegraph model predicts that these parameters should collapse as a function of the scaling parameter

(17)$$\alpha = \displaystyle{{\varepsilon g} \over a}$$

where a quantifies the average alignment of the chain backbone with the channel axis. Remarkably, the limits of large α (narrow channels) and small α (wide channels), respectively, correspond to the parameter ξ for the backfolded Odijk regime in Eq. (14) and the scaling variable (wl _p/D ²) in the extended de Gennes regime. As a result, the limits of the telegraph model are the backfolded Odijk and extended de Gennes regime (Werner et al., Reference Werner, Cheong, Gupta, Dorfman and Mehlig2017).

Fig. 4. Schematic illustration of the weakly correlated telegraph model (Werner et al., Reference Werner, Cheong, Gupta, Dorfman and Mehlig2017). The chain is replaced by a particle that moves at a constant velocity between lattice sites with random changes in direction. When a lattice site is revisited, there is an energy penalty.

In addition to proposing the new scaling variable in Eq. (17), Werner et al. (Reference Werner, Cheong, Gupta, Dorfman and Mehlig2017) showed how the confined polymer problem maps to the correlated telegraph model. Explicitly, the various quantities appearing in the telegraph model can be computed from the statistics of an ideal semiflexible chain, i.e. when the only excluded volume interactions are between the polymer and the wall, and then the telegraph model provides a way to combine those statistics to predict the behavior of a real polymer. Figure 5 shows that the resulting predictions of the telegraph model are in almost exact agreement with three-dimensional simulation data for a confined wormlike chain (Werner et al., Reference Werner, Cheong, Gupta, Dorfman and Mehlig2017). The only disagreement between theory and simulation arises deep into the backfolded Odijk regime at large α. Here, where hairpin turns are rare, the dominant contribution to the variance becomes the fluctuations in the alignment of the segments with the channel walls, which are not included in the telegraph model. Simply adding the variance of the telegraph model to the result in Eq. (12) for these alignment fluctuations restores the agreement between the theory and simulation. The collapse of the experimental data to the theory in Fig. 5 is also satisfying, especially in light of the possible sources of systematic errors that have been identified in the context of experimental work (Nyberg et al., Reference Nyberg, Persson, Akerman and Westerlund2013; Gupta et al., Reference Gupta, Miller, Muralidhar, Mahshid, Reisner and Dorfman2015; Iarko et al., Reference Iarko, Werner, Nyberg, Muller, Fritzsche, Ambjornsson, Beech, Tegenfeldt, Mehlig, Westerlund and Mehlig2015; Schotzinger et al., Reference Schotzinger, Menard and Ramsey2020).

Fig. 5. The telegraph model compared to simulations (a, b) and experiments (c, d), respectively. The dashed lines in (b) (referring to Eq. (10) in the original paper) is the sum of the variance predicted by the telegraph model and alignment fluctuations from Odijk's theory (Burkhardt et al., Reference Burkhardt, Yang and Gompper2010). The pertinent scaling laws are described by Werner et al., (Reference Werner, Cheong, Gupta, Dorfman and Mehlig2017). Experimental data: ▿ (Reisner et al., Reference Reisner, Morton, Riehn, Wang, Yu, Rosen, Sturm, Chou, Frey and Austin2005), □ (Reisner et al., Reference Reisner, Beech, Larsen, Flyvbjerg, Kristensen and Tegenfeldt2007), ■ (Thamdrup et al., Reference Thamdrup, Klukowska and Kristensen2008), ○ (Zhang et al., Reference Zhang, Zhang, Kan and Maarel2008), • (Utko et al., Reference Utko, Persson, Kristensen and Larsen2011), ▵ (Kim et al., Reference Kim, Kim, Kounovsky, Chang, Jung, Depablo, Jo and Schwartz2011), ▴ (Werner et al., Reference Werner, Persson, Westerlund, Tegenfeldt and Mehlig2012), ▾ (Gupta et al., Reference Gupta, Sheats, Muralidhar, Miller, Huang, Mahshid, Dorfman and Reisner2014), ⋄ (Reinhart et al., Reference Reinhart, Reifenberger, Gupta, Muralidhar, Sheats, Cao and Dorfman2015), ♦ (Gupta et al., Reference Gupta, Miller, Muralidhar, Mahshid, Reisner and Dorfman2015), ⬠ (Iarko et al., Reference Iarko, Werner, Nyberg, Muller, Fritzsche, Ambjornsson, Beech, Tegenfeldt, Mehlig, Westerlund and Mehlig2015), and ⬟ (Alizadehheidari et al., Reference Alizadehheidari, Werner, Noble, Reiter-Schad, Nyberg, Fritzsche, Mehlig, Tegenfeldt, Ambjörnsson, Persson and Westerlund2015). Reproduced from Werner et al. (Reference Werner, Cheong, Gupta, Dorfman and Mehlig2017), © Werner et al. (Reference Werner, Cheong, Gupta, Dorfman and Mehlig2017) under Creative Commons CC BY 4.0.

Chen (Reference Chen2018) arrived at a similar scaling result to Eq. (17) using the self-consistent field theory solution for the wormlike chain propagator. Chen's theoretical approach is more robust and allows considering polymers that are much stiffer than DNA and inaccessible by the simulations needed to compute the parameters appearing in the telegraph model.

The telegraph model can also be solved exactly (Odman et al., Reference Odman, Werner, Dorfman, Doering and Mehlig2018) in the limit of relevance to genome mapping, where the channel cross-section is close to the DNA persistence length, providing a formula for predicting not only the mean span and its variance, but also the distribution for the chain extension. Simulations of semiflexible polymers by Bhandari and Dorfman (Reference Bhandari and Dorfman2019), including situations that mimic DNA-based experiments, showed that the asymptotic solution of Odman et al. (Reference Odman, Werner, Dorfman, Doering and Mehlig2018) is in quantitative agreement with simulations. Experimental data for the DNA extension distribution obtained in genome mapping experiments (Reinhart et al., Reference Reinhart, Reifenberger, Gupta, Muralidhar, Sheats, Cao and Dorfman2015; Sheats et al., Reference Sheats, Reifenberger, Cao and Dorfman2015), however, tend to exhibit increasingly worse agreement with the asymptotic solution to the telegraph model as the channel size decreases. This behavior is in contrast to the expectations of the theory of Odman et al. (Reference Odman, Werner, Dorfman, Doering and Mehlig2018), that infer improved agreement as the channel size decreases. Subsequent experiments and theory by Chuang et al. (Reference Chuang, Reifenberger, Bhandari and Dorfman2019) suggest that this disagreement is due to the simplified approach to DNA–wall electrostatic interactions used to model DNA, not an inherent problem in the telegraph model itself.

Effect of channel geometry

The discussion thus far has focused primarily on square nanochannels of width D. From a theoretical perspective, there are only minor differences between a square channel and a circular channel of diameter D that arise from the entropic depletion of the DNA from the corners of the square channel (Odijk, Reference Odijk2006). While simple to analyze from a theoretical perspective, these geometries are challenging to fabricate in practice. The most commonly encountered situation is rectangular nanochannels with a width D ₁ and a height D ₂.

The first question to address is whether a rectangular cross-section is actually a nanochannel in the sense that we have discussed so far, or whether it is a slit. To answer this question, let us assume that D ₁ > D ₂, which is the easier case to fabricate, and recall what would happen in the case D ₁ → ∞. This situation corresponds to a nanoslit of height D ₂. Daoud and De Gennes (Reference Daoud and De Gennes1977) used scaling theory to demonstrate that the in-plane radius of gyration of a polymer confined to a slit consists of a two-dimensional, self-avoiding random walk of blobs. For a semiflexible polymer, Eq. (3) for a polymer in free solution becomes (Hsieh et al., Reference Hsieh, Balducci and Doyle2008)

(18)$$R_{{\rm slit}}^2 \sim L^{3/2}\left({\displaystyle{{l_pw} \over {D_2}}} \right)^{1/2}$$

when that polymer is confined to a slit. Provided that R _slit ≫ D ₂, then the system is a nanochannel because the DNA configuration will be influenced by the presence of the side wall. The latter inequality thus defines the cross-over between a nanoslit and a nanochannel.

Once we have confirmed that the system indeed corresponds to a nanochannel, we need to understand how the results for confinement discussed earlier are affected by the aspect ratio. To a first approximation, the theories for DNA confinement emerge from the random walk statistics of the confined chain (Chen, Reference Chen2016). In the random walk statistics, the diffusion in different directions is decoupled. As a result, one can define the equivalent square channel size as D ² = D ₁D ₂. For aspect ratios that are not too far from unity, this approximation is sufficient. However, as the aspect ratio grows, additional complications arise because the way in which the confinement causes the DNA to extend along the channel axis depends on the channel size. This effect becomes particularly important for funnel-shaped nanochannels (Persson et al., Reference Persson, Utko, Reisner, Larsen and Kristensen2009). If the length scales D ₁ and D ₂ correspond to different regimes of confinement, a ‘regime mixing’ effect occurs that complicates the analysis (Gupta et al., Reference Gupta, Sheats, Muralidhar, Miller, Huang, Mahshid, Dorfman and Reisner2014). It is possible to construct scaling theories that account for the regime mixing through the use of superblobs (Werner and Mehlig, Reference Werner and Mehlig2015). However, the resulting theory now requires satisfying two strong inequalities, one for each of the channel dimensions, a situation that is unlikely to be realized in experiments. For rectangular channels, the most powerful approach to understanding the chain statistics is simulation (Gupta et al., Reference Gupta, Sheats, Muralidhar, Miller, Huang, Mahshid, Dorfman and Reisner2014; Muralidhar et al., Reference Muralidhar, Quevillon and Dorfman2016).

In addition to rectangular channels, there is also a small body of literature on simulations of DNA confinement in triangular nanochannels (Manneschi et al., Reference Manneschi, Angeli, Ala-Nissila, Repetto, Firpo and Valbusa2013; Reinhart et al., Reference Reinhart, Tree and Dorfman2013; Han et al., Reference Han, Kim, Matsuoka, Thouless and Takayama2016), which are also relatively easy to fabricate by anisotropic etching of silicon. The entropic depletion effects of acute corners can be very significant, and the collapse of the extension data onto a universal curve is not straightforward (Reinhart et al., Reference Reinhart, Tree and Dorfman2013). Ultimately, triangular nanochannels have found interesting applications as collapsible systems for strongly extending DNA (Huh et al., Reference Huh, Mills, Zhu, Burns, Thouless and Takayama2007) and chromatin (Matsuoka et al., Reference Matsuoka, Kim, Huang, Douville, Thouless and Takayama2012) rather than as a system for fundamental studies.

Alternative fabrication

For many of the applications foreseen for nanofluidic devices, in particular in diagnostics, disposable chips produced at low cost are desirable. Several different strategies have been proposed for making chips using less demanding techniques and at higher throughput, compared to the traditionally used state-of-the-art nanofabrication processes. Nanofluidic chips for DNA analysis based on thermoplastic polymers can be fabricated by injection molding (Utko et al., Reference Utko, Persson, Kristensen and Larsen2011) or nanoimprinting (Wu et al., Reference Wu, Chantiwas, Amirsadeghi, Soper and Park2011; Uba et al., Reference Uba, Pullagurla, Sirasunthorn, Wu, Park, Chantiwas, Cho, Shin and Soper2015; Esmek et al., Reference Esmek, Bayat, Pérez-Willard, Volkenandt, Blick and Fernandez-Cuesta2019). Devices replicated in PDMS (van Kan et al., Reference van Kan, Zhang, Perumal Malar and van der Maarel2012) from a nanofabricated master is another approach to increase fabrication throughput and decrease cost. Park et al. (Reference Park, Oh and Cho2018) reported how near-field electrospinning can be used to create nanofluidic channels with a circular cross-section in PDMS, suitable for biological applications.

Tapered channels

By varying the channel dimensions along the channel length, it is possible to study the same individual, single DNA molecule at different degrees of confinement (Fig. 8a, b). In analogy to force spectroscopy, Persson et al. coined the concept confinement spectroscopy for probing single DNA molecules in tapered nanochannels (Persson et al., Reference Persson, Utko, Reisner, Larsen and Kristensen2009). Similar funnel-shaped channels were used by Frykholm et al. (Reference Frykholm, Alizadehheidari, Fritzsche, Wigenius, Modesti, Persson and Westerlund2014) and Fornander et al. (Reference Fornander, Frykholm, Fritzsche, Araya, Nevin, Werner, Cakir, Persson, Garcin, Beuning, Mehlig, Modesti and Westerlund2016), when studying the nucleoprotein filaments formed by RecA and Rad51, respectively, on DNA. Channels where the width is varied in discrete steps, rather than continuously tapered, have also been used, for example by Werner et al. (Reference Werner, Jain, Muralidhar, Frykholm, St Clere Smithe, Fritzsche, Westerlund, Dorfman and Mehlig2018) and Gupta et al. (Reference Gupta, Miller, Muralidhar, Mahshid, Reisner and Dorfman2015). The advantage of using discrete steps is that the DNA is not subjected to an entropic force that pushes it toward the larger end of the funnel, allowing equilibrium measurements to be performed at the cost of only doing so at a discrete number of channel dimensions.

Fig. 8. Schematic illustration of chip designs tailored for different applications. (a–b) Varying channel dimensions, in a tapered or stepwise fashion, along the channel length enables studying the same individual, single DNA molecule at different degrees of confinement. (c) A chip design with perpendicular channel arrays enables diffusion driven buffer exchange. (d) Aligning a shallow slit on top of and orthogonal to reaction chambers, flanked by nanofluidic channels, enables active buffer exchange in the local environment of entrapped single DNA molecules. (e) Physical barriers that gradually increase the confinement can provide pre-stretching of DNA and reduce the thermodynamic cost of entering genomic sized DNA into nanochannels. (f) A meandering nanochannel design enables visualization of ultralong DNA molecules in a single field of view. The sketches are not drawn to scale.

Devices for dynamic studies

The nanochannel device can be designed to add functionalities to it, such as the possibility to change the local environment around the DNA molecule under study, i.e. exchange buffers or add reagents to it. Such a set-up enables studies of, for example, DNA–protein interactions in real time.

A chip design presented by Zhang et al. features two arrays of parallel nanochannels in a perpendicular configuration (Zhang et al., Reference Zhang, Jiang, Liu, Doyle, van Kan and van der Maarel2013b) (Fig. 8c). DNA is introduced into the device via electrophoresis through the array of wider (250 × 200 nm²) nanochannels, whereas buffer can be exchanged via diffusion through the intersecting array of narrower (150 × 200 nm²) nanochannels in the perpendicular direction. As a proof of principle, compaction of T4-DNA upon addition of protamine and unpacking of pre-compacted T4-DNA by increasing salt concentration were investigated, demonstrating the functionality of the device (Fig. 9a). Subsequently, the same device was used to study the interaction of the bacterial protein H-NS with DNA (Zhang et al., Reference Zhang, Guttula, Liu, Malar, Ng, Dai, Doyle, van Kan and van der Maarel2013a).

Fig. 9. (a) A nanofluidic device featuring two arrays of nanochannels oriented in a perpendicular configuration enables insertion of biomolecules through the channels in one direction and diffusion-driven buffer exchange through the intersecting, narrower, channels. The functionality was demonstrated by compaction of T4–DNA upon exposure to protamine (1 (△), 3 (○) and 5 (▽) μM, respectively; inset shows fluorescence images of T4-DNA upon exposure to 5 μM protamine). Reproduced with permission from Zhang et al. (Reference Zhang, Jiang, Liu, Doyle, van Kan and van der Maarel2013b), © 2013 The Royal Society of Chemistry. (b) Demonstration of active exchange of the local environment in a nanofluidic reaction chamber by sequential addition of EtBr and SDS to YOYO-1 stained λ-DNA. Kymographs showing emission from YOYO-1 (top) and EtBr (center), respectively, and corresponding fluorescence intensity profiles (bottom) for YOYO-1 (green) and EtBr (red). Scale bar corresponds to 5 μm. Reproduced from Öz et al. (Reference Öz, Sriram and Westerlund2019), © Öz et al. (Reference Öz, Sriram and Westerlund2019) under Creative Commons CC BY-NC 3.0. (c) Schematic illustration of the CLiC principle (top): DNA molecules in the solution are initially unconfined, as the height of the chamber is of microscale dimensions (left). Upon lowering the push-lens the chamber height is reduced to nanoscale dimensions and the DNA molecules are aligned to the nanochannels by confinement (right). Two series of fluorescence images showing the extension of λ-DNA as confinement increases while the push-lens is lowered. Center panel: channel width 27 nm and time between frames 364 ms; bottom panel: channel width 50 nm and time between frames 910 ms. Reproduced from (Berard et al., Reference Berard, Michaud, Mahshid, Ahamed, McFaul, Leith, Bérubé, Sladek, Reisner and Leslie2014), © (Berard et al., Reference Berard, Michaud, Mahshid, Ahamed, McFaul, Leith, Bérubé, Sladek, Reisner and Leslie2014).

Öz et al. (Reference Öz, Sriram and Westerlund2019) designed a device enabling active exchange of the local environment within the nanofluidic channel (Fig. 8d). The design includes an array of nanochannels (100 × 100 nm²), with a central stretch of slightly larger dimensions (300 × 130 nm²), creating an entropically controlled reaction chamber where single DNA molecules can be confined and stretched. Across the reaction chambers, on top of the array and in an orthogonal fashion, runs a shallow nano-slit (30 μm × 30 nm), through which analytes actively can be introduced via pressure-driven flow to exchange the local buffer environment of the entrapped DNA molecule. The authors demonstrated the functionality of the device by a series of experiments, where analytes, such as SDS, spermidine, ethidium bromide and DNase I, were added to confined DNA molecules and the resulting response could be monitored in situ (Fig. 9b). The same device was later used when studying conformational changes in large DNA upon interaction with DNA-binding proteins (Sharma et al., Reference Sharma, Sriram, Holmstrom and Westerlund2020; Jiang et al., Reference Jiang, Humbert, S, Rouzina, Mely and Westerlund2021a).

Handling ultralong DNA

In the use of nanofluidic channels for ODM applications, discussed in detail below, the size of the DNA of interest can pose experimental challenges. Introducing DNA of genomic size to a nanosized channel is associated with a significant entropic barrier, and coils of large DNA molecules tend to entangle and clog the channel entrance. An approach to promote DNA loading into nanochannels and facilitate DNA entering in an unfolded state with one end first is to include an entropic gradient in the chip design. Physical barriers such as pillar structures, that gradually increase the confinement, provide pre-stretching of the DNA and reduce the thermodynamic cost (Cao et al., Reference Cao, Tegenfeldt, Austin and Chou2002a; Wang et al., Reference Wang, Bruce, Duch, Patel, Smith, Astier, Wunsch, Meshram, Galan, Scerbo, Pereira, Wang, Colgan, Lin and Stolovitzky2015). Lam et al. (Reference Lam, Hastie, Lin, Ehrlich, Das, Austin, Deshpande, Cao, Nagarajan, Xiao and Kwok2012) applied this approach to genome mapping in a nanofluidic device that contained a region with channels of intermediate width and physical barriers between the feeding microfluidic channel and the nanochannel array (Fig. 8e).

Another concern to address when long, genomic DNA is confined to nanofluidic channels and visualized using fluorescence microscopy is that the size of the DNA molecule might exceed the field of view of the camera at sufficient resolution. By designing the channels in a meandering fashion (Fig. 8f) Freitag et al. circumvented this issue (Freitag et al., Reference Freitag, Noble, Fritzsche, Persson, Reiter-Schad, Nilsson, Graneli, Ambjornsson, Mir and Tegenfeldt2015). In this study, a single molecule of an intact 5.7 Mbp Schizosaccharomyces pombe (S. pombe) chromosome was stretched, at an extension of 50% of its contour length to 1 mm, in a meandering nanochannel and visualized in a single field of view.

Chromatin

Eukaryotic chromatin is comprised of DNA wrapped up on histone protein complexes, forming a densely packed structure that enables fitting the meter-long DNA in each cell into the cell nucleus. The structure of chromatin DNA plays a crucial role in gene expression and silencing and is therefore subject to heavy regulation, to a large extent governed by epigenetic modifications on the histone tails. To study linearized chromatin fibers and map genetic and/or epigenetic modifications is thus of great interest and nanofluidic channels could potentially be well suited for this purpose.

Streng et al. reported the first study of chromatin stretched in nanochannels (Streng et al., Reference Streng, Lim, Pan, Karpusenka and Riehn2009), where the chromatin used was reconstituted from a commercially available, unfractionated whole histone mixture on blunt-ended λ-DNA that was fluorescently labeled using YOYO-1. The extension of chromatin (in 80 × 80 nm² channels) was compared to that of bare DNA and chromatin was found to be 2.5 times more compact. However, the authors discuss that the chromatin was probably not in the form of condensed 30 nm fibers but rather a disordered 10 nm fiber with heterogeneous linker lengths, and physical parameters for this non-homogeneous chromatin could be given as rough estimates only.

More recently, Basak et al. (Reference Basak, Rosencrans, Yadav, Yan, Berezhnoy, Chen, van Kan, Nordenskiöld, Zinchenko and van der Maarel2020) investigated the compaction, dynamics, and internal motion of nanoconfined chromatin fibers. They used over-expressed, refolded, and purified histone octamers to reconstitute chromatin on T4-DNA, and nanofluidic devices with two different degrees of confinement (50 × 70 nm² and 120 × 130 nm², respectively). The compaction of DNA was followed by monitoring the extension of bare T4-DNA and chromatin fibers formed with an octamer loading of 50 or 100%, respectively, when confined to nanochannels (Fig. 13a). The reconstituted chromatin fibers were found to not achieve a fully compacted state and fluctuations between relatively more and less open configurations were observed. The internal motion of DNA was further investigated through analysis of fluorescence correlation, from which it was deduced that this motion is governed mainly by the dynamics of uncompressed linker DNA.

Fig. 13. (a) Naked T4-DNA and chromatin fibers, reconstituted with T4-DNA and two different loadings of recombinant histone octamers, stretched in nanofluidic channels. DNA is visualized by staining with YOYO-1. Chromatin fibers appear as blobs connected by strands of uncompressed DNA, and these blobs were observed to continuously reform so that the fibers fluctuate between relatively more and less open configurations. Reproduced with permission from Basak et al. (Reference Basak, Rosencrans, Yadav, Yan, Berezhnoy, Chen, van Kan, Nordenskiöld, Zinchenko and van der Maarel2020), © 2020 Biophysical Society. (b) Real-time visualization of DNA compaction upon addition of the NC protein. (A) Kymograph of YOYO-1 stained λ-DNA compacted and finally condensed by NC, starting in the center of the molecule (white arrow). (B) 3D surface plot showing the formation of the local condensate, at the time point corresponding to the dashed square in (A), with emission intensity color coded. (C) Normalized emission intensity profiles of DNA at the corresponding time points indicated in (A). (D) As in (A) but with the local condensate forming at the end of the DNA. The horizontal scale bars are 2 μm. Reproduced from Jiang et al. (Reference Jiang, Humbert, S, Rouzina, Mely and Westerlund2021a), © Jiang et al., (Reference Jiang, Humbert, S, Rouzina, Mely and Westerlund2021a) under Creative Commons CC BY-NC 4.0.

Identification of histone tail modifications using fluorescent antibodies on chromatin stretched in nanochannels was demonstrated by Lim et al. (Reference Lim, Karpusenko, Blumers, Streng and Riehn2013). In this study, chromatin was reconstituted from calf thymus, HeLa core, and chicken erythrocyte histones and antibodies detecting trimethylation of lysine 4 and acetylation of lysine 9, respectively, on histone 3 (H3) was used to monitor histone modifications. The ratio between these modifications varies among chromatin origins and discrimination between the three histone samples was demonstrated. Channels wider than traditionally used for DNA analysis (200 × 200 nm²), resulting in moderate stretching, were used in this study to avoid stripping of histones from the DNA and antibody aggregation.

Another approach to overcome the risk of stripping the histones off the DNA was presented by Matsuoka et al. (Reference Matsuoka, Kim, Huang, Douville, Thouless and Takayama2012), who used an elastomeric nanochannel device where the channel dimensions can be adjusted post introduction of the chromatin. The chromatin in this study was isolated from HeLa cells and by co-staining with DAPI and antibodies against methylated histone H3 and acetylated histone H4, respectively, the authors were able to visualize chromatin condensation and localization of epigenetic marks on the linearized chromatin. The study points out several advantages of using nanochannels for chromatin analysis: no extra chemistry is required to stretch the chromatin, chromatin extracted from cells can be studied, and multicolor staining enables analysis of several epigenetic markers simultaneously. The latter, together with the single chromatin approach, is crucial for rare samples.

Bacterial nucleoid-associated proteins

The bacterial chromosome is organized into a structure called the nucleoid, where the DNA forms nucleoprotein complexes with several proteins referred to as nucleoid-associated proteins (NAPs) (Azam and Ishihama, Reference Azam and Ishihama1999). Among the most abundant and universally conserved NAPs are the heat-stable nucleoid structuring protein (H-NS) and the heat unstable nucleoid structuring protein (HU) and they have both been subject to studies using nanofluidic channels.

Zhang et al. (Reference Zhang, Guttula, Liu, Malar, Ng, Dai, Doyle, van Kan and van der Maarel2013a) studied the effect of H-NS on DNA conformation using two different set-ups of nanofluidic channels. In steady-state experiments, an array of nanochannels was used to visualize the extension of single DNA molecules, pre-incubated with H-NS, whereas a cross-channel device (Zhang et al., Reference Zhang, Jiang, Liu, Doyle, van Kan and van der Maarel2013b), described in section ‘Devices for dynamic studies’, enabled monitoring the time-dependent response of DNA upon exposure to H-NS. DNA was found to elongate or condense, depending on protein concentration, ionic strength, and presence of divalent ions (Mg²⁺), and H-NS mediated bridging of DNA was observed.

Later, a similar study on HU was presented (Guttula et al., Reference Guttula, Liu, van Kan, Arluison and van der Maarel2018). Contrary to the findings for H-NS, nanoconfined DNA that had been pre-incubated with HU showed a decrease in extension with increasing protein concentrations, irrespective of ionic strength and buffer composition (presence of Mg²⁺ ions). Over-threshold concentrations of protein caused DNA to collapse into a condensed conformation.

Another DNA-binding protein within the bacterial nucleoid that has been studied using nanofluidic channels is Hfq, a phylogenetically conserved and abundant bacterial protein with multiple regulatory functions related to nucleic acids metabolism. Jiang et al. (Reference Jiang, Zhang, Guttula, Liu, van Kan, Lavelle, Kubiak, Malabirade, Lapp, Arluison and van der Maarel2015) observed compaction of DNA in the presence of wild-type, full length Hfq. The decrease in molecular extension was, just as for H-NS, explained by the propensity of Hfq to bridge distal DNA segments and at over-threshold concentrations of Hfq the DNA was found to collapse into a condensed form. In a follow-up study, truncated Hfq proteins, representing either the proposed DNA-binding C-terminal domain (Hfq-CTR) or the N-terminal region (Hfq-NTR), were investigated (Malabirade et al., Reference Malabirade, Jiang, Kubiak, Diaz-Mendoza, Liu, van Kan, Berret, Arluison and van der Maarel2017). As a result, a proposed model for DNA binding and bridging by Hfq, where the C-terminal of the protein plays a pivotal role, with possible implications for protein-binding-related gene regulation, was presented. Recently, the effect of intramolecular DNA motion on the mobility of Hfq was investigated using a nanofluidic set-up, demonstrating a dynamic coupling between confinement-induced slowing of DNA internal motion and enhanced protein mobility (Yadav et al., Reference Yadav, Basak, Yan, van Kan, Arluison and van der Maarel2020). Subsequently, a mutant form of Hfq, lacking the C-terminal domain and having a lower DNA binding affinity, was found to have a significantly higher mobility on DNA compared to the wild-type protein (Tan et al., Reference Tan, Basak, Yadav, van Kan, Arluison and van der Maarel2022). The mutant also demonstrated a reverse dependence on the internal motion of DNA, showing reduced protein mobility with slower internal motion.

Other DNA compacting proteins

Condensation and regulation of DNA accessibility are key activities also in virus life cycles. The nucleocapsid protein NC, alone and as part of the polyprotein Gag, from human immunodeficiency virus type 1 (HIV-1) has been investigated using nanochannels in two studies by Jiang et al. (Reference Jiang, Humbert, Sriram, Lequeu, Lin, Mely and Westerlund2019, Reference Jiang, Humbert, S, Rouzina, Mely and Westerlund2021a). Protein concentration-dependent compaction of nanoconfined dsDNA was demonstrated, and Gag, which has a higher binding affinity to DNA than NC alone, was shown to condense DNA at a much lower concentration than NC (Jiang et al., Reference Jiang, Humbert, Sriram, Lequeu, Lin, Mely and Westerlund2019). In the same study, the ability of NC to act as a chaperone and promote annealing of complementary sequences was investigated. By using λ-DNA, that has 12 bp single-stranded overhangs in each end, concatemerization and circularization in the presence of NC could be observed. Recently, a further investigation of the dynamics and kinetics of dsDNA compaction induced by NC was presented (Jiang et al., Reference Jiang, Humbert, S, Rouzina, Mely and Westerlund2021a). A dynamic device, first described by Öz et al. (Reference Öz, Sriram and Westerlund2019) and discussed in section ‘Devices for dynamic studies', enabled real-time observation of single DNA molecules upon addition of NC protein and the dynamics of the condensation process could thereby be followed, from locally compacted regions on the DNA to the fully condensed state (Fig. 13b). The NC-promoted annealing of complementary, single-stranded overhangs of DNA could also be visualized in real time in the dynamic device and the authors concluded that the chaperone activity is not directly correlated with the DNA condensation propensity of NC.

Another virus nucleocapsid protein studied in nanofluidic channels is the hepatitis C virus core protein (HCVcp) (Sharma et al., Reference Sharma, Sriram, Holmstrom and Westerlund2020). HCVcp-induced compaction of DNA was monitored in real time, and repeated compaction and relaxation of DNA could be visualized by iterative flushing of the reaction chamber with either HCVcp (initiating compaction) or Proteinase K (releasing the DNA by degradation of the protein). It was observed that compaction preferentially starts from the ends of linear DNA, whereas for a circular substrate compaction was initiated at random positions along the contour. These observations highlight two major advantages of nanofluidic channels, over many other single-molecule techniques: as the DNA is free in solution in the channels, without any tethers, events occurring at the ends of the molecule and on circular DNA, lacking ends that make it difficult to tether, can be studied.

Stretching and visualizing single DNA–protein complexes in nanofluidic channels can provide complementary information to confirm or add detail to structural predictions from X-ray crystallography and/or models of biomolecular functions. Frykholm et al. (Reference Frykholm, Berntsson, Claesson, De Battice, Odegrip, Stenmark and Westerlund2016) studied the compaction of nanoconfined DNA by the bacteriophage protein Cox, a multifunctional transcriptional regulator in P2-like bacteriophages. Cox forms oligomeric filaments and it has been proposed from the X-ray structure that DNA can be wrapped around these filaments, in a manner similar to how histones condense DNA in eukaryotic cells. The nanofluidic experiments confirmed that Cox compacts DNA and that the binding is highly cooperative, in agreement with the postulated model. By monitoring the DNA extension at different channel widths, it was illustrated how the physical properties of the complex at low protein loads resemble those of naked DNA, whereas they are governed by the stiffer Cox-filament at higher protein coverage.

In a multimodal study by Schmitt et al. (Reference Schmitt, Jiang, Camacho, Jonna, Hofer, Westerlund, Christie and Berntsson2018) the structure, properties, and functions of PrgB, a surface adhesin of Enterococcus faecalis, were investigated. Nanofluidic channels were used to analyze PrgB binding to long DNA fragments. Compaction of DNA was observed, with full condensation of DNA at over-threshold protein concentration and deduced cooperative binding at lower concentrations. The investigation also indicated a higher affinity of PrgB for ends of DNA substrates or, alternatively, for the single-stranded overhangs present on the DNA substrates used.