One-parameter scaling theory for DNA extension in a nanochannel

TL;DR

This paper introduces a one-parameter scaling theory based on a telegraph process that accurately models DNA extension in nanochannels, resolving discrepancies between experiments and existing theories.

Contribution

It develops a novel theoretical framework that unifies experimental and simulation data for DNA confinement, improving understanding of polymer behavior in nanochannels.

Findings

The theory collapses diverse data onto a single master curve.

It explains the mechanisms behind DNA extension in nanochannels.

The model aligns well with experimental and simulation results.

Abstract

Experiments measuring DNA extension in nanochannels are at odds with even the most basic predictions of current scaling arguments for the conformations of confined semiflexible polymers such as DNA. We show that a theory based on a weakly self-avoiding, one-dimensional "telegraph" process collapses experimental data and simulation results onto a single master curve throughout the experimentally relevant region of parameter space and explains the mechanisms at play.

Figures

Equations

$$P[x(s)]\propto P_{\mathrm{ideal}}[x(s)]\mathcal{A}[x(s)].$$

$$⟨v_x(s)v_x(0)⟩=a^2\exp (-s/g).$$

$$\text{mbox}⟨v(t)v(0)⟩=v_0^2\exp (-2rt).$$

$$\mathscr{A}[x(s)]\propto{\rm exp}\big{[}-\tfrac{{\varepsilon}}{2}\textstyle\int\!{\rm d}x\,{\mathscr{L}}^{2}(x)\big{]}\,.$$

$$\epsilon =⟨\delta (y-y^{\prime })\delta (z-z^{\prime })v_{\mathrm{ex}}⟩/{\ell }^2.$$

$$\alpha \equiv \epsilon /(2v_{0}r).$$

$$\alpha =\epsilon g/a.$$

$$X/(La)=f_X(\alpha )\text{mbox}and{\sigma }^2/(Lg{a}^2)=f_{\sigma }(\alpha ).$$

$$f_X(\alpha )$$

$$f_{\sigma }(\alpha )$$

$${\sigma }^2=d_{\sigma }(Lg{a}^2){\alpha }^{-3}+{\sigma }_{\mathrm{Odijk}}^2\text{mbox}as\alpha \to \infty .$$

$$\alpha =Cgw(D^5{\ell }_{\mathrm{P}}{)}^{-1/3}=C\xi (D\ll {\ell }_{\mathrm{P}}),$$

$$\alpha =9\sqrt{3}\pi w{\ell }_{\mathrm{P}}/(8D^2)\ll 1({\ell }_{\mathrm{P}}\ll D\ll {\ell }_{\mathrm{P}}^2/w).$$

Full text

As the carrier of genetic information, DNA plays a key role in biology. At the same time, recent advancements in fluorescence microscopy allow DNA to serve as a model polymer for investigating fundamental questions in polymer physics Shaqfeh (2005); Latinwo and Schroeder (2011). Nowhere is this dual importance more apparent than in the problem of DNA confinement in a nanochannel Reisner et al. (2012); Dorfman et al. (2013); Dai et al. (2016). When the radius of gyration of the DNA molecule is larger than the channel width, it extends along the channel. This stretching lies at the heart of genome mapping in nanochannels Lam et al. (2012). Here the stretched DNA molecules, usually greater than 150 kilobasepairs in length, contain fluorescent markers that reveal sequence-specific information with kilobasepair resolution. This new method serves as a complement to next-generation de novo DNA sequencing Jo et al. (2007); Lam et al. (2012); Michaeli and Ebenstein (2012).

Polymer confinement has been investigated for four decades, starting with the scaling arguments of Daoud and de Gennes Daoud and De Gennes (1977). Yet there is to date no microscopic theory explaining the experimental data of recent genome-mapping experiments Jo et al. (2007); Lam et al. (2012); Michaeli and Ebenstein (2012); Reinhart et al. (2015) in narrow nanochannels. The difficulty is that the channels are too wide to apply scaling arguments derived for strong confinement Odijk (2008); Muralidhar et al. (2014), yet too narrow for the scaling arguments and theory Odijk (2008); Wang et al. (2011); Dai et al. (2014); Werner and Mehlig (2014, 2015) in wider channels to hold.

The challenge in developing a theory for the extension of nanoconfined DNA arises from its semiflexible nature. Semiflexible polymers are characterized by three length scales: the contour length $L$, the persistence length $\ell_{\rm P}$ quantifying the stiffness of the chain, and the effective width $w$ that appears in the Onsager excluded volume Onsager (1949). For polyelectrolytes such as DNA, both the persistence length Odijk (1977); Skolnick and Fixman (1977); Dobrynin (2005); Trizac and Shen (2016) and the effective polymer width Stigter (1977) depend on electrostatic interactions. Recent experiments are often conducted in high ionic-strength buffers. In this case $\ell_{\rm P}$ is approximately $50\,$nm Bustamante et al. (1994) while $w$ is around $5\,$nm Hsieh et al. (2008), and thus $w\ll\ell_{\rm P}$. This inequality emphasizes the intrinsic difficulty of describing DNA in a wide range of situations. DNA is considerably stiffer than typical synthetic polymers, yet the number of persistence lengths $L/\ell_{\rm P}$ in genomic DNA samples is large. Any theory for the conformational statistics of channel-confined DNA must account for both the local stiffness of the polymer and excluded-volume interactions. This is a formidable challenge. Matters are further complicated by the fact that most recent genome-mapping experiments are performed in nanochannels that are about $50\,$nm wide Jo et al. (2007); Lam et al. (2012); Michaeli and Ebenstein (2012); Reinhart et al. (2015). Essentially all experiments involving DNA (Fig. 1) take place in channel sizes $D$ of the order of $\ell_{\rm P}$ and do not satisfy the criterion $D\gg\ell_{\rm P}^{2}/w$ required for the scaling arguments of Ref. Daoud and De Gennes (1977) to apply.

There is no microscopic theory for the extension of confined DNA for $D<\ell_{\rm P}^{2}/w$, despite numerous attempts Zhang et al. (2008); Cifra (2009); Tree et al. (2013a). Scaling arguments Wang et al. (2011); Dai et al. (2014); Werner and Mehlig (2015) following Refs. Daoud and De Gennes (1977); Odijk (2008) yield the most useful description. If $D\gg\ell_{\rm P}$ they suggest that the extension $X$ scales as $X\sim D^{-2/3}$. The problem is that the theory fails when $D\approx\ell_{\rm P}$, and as a result it proves to be a poor description of most recent DNA experiments in nanochannels. The earliest experiments Reisner et al. (2005), for instance, reported a much larger exponent $X\sim D^{-0.85}$, and subsequent studies Persson et al. (2009); Gupta et al. (2014, 2015) continue to report exponents exceeding the theoretical prediction.

We take a different approach in this Letter. We show that the DNA-confinement problem for $w\ll\ell_{\rm P}$ and $D\ll\ell_{\rm P}^{2}/w$ maps to the simple one-dimensional telegraph process in Fig. 2, describing the correlated walk of a particle moving with velocity $v_{0}$ along the channel axis. The velocity changes sign at rate $r$, creating hairpin configurations in the particle path. The process lasts from $t=0$ to $t=T$. When the particle revisits a position it has previously visited, it incurs a penalty ${\varepsilon}$. We show that this model collapses experimental and simulation data for the extension throughout the experimentally relevant parameter range onto a universal master curve as a function of a new scaling variable, $\alpha$, that measures the combined effects of confinement, local stiffness, and self avoidance.

We start by considering narrow channels, $D\ll\ell_{\rm P}$, and later extend the arguments to channel widths up to $\ell_{\rm P}^{2}/w$. Since we want to compute the extension of the DNA molecule along the channel axis, it suffices to consider the projection of the three-dimensional DNA configurations \mbox{\boldmathx}(s) to the channel axis $x$. Here $s$ is the contour-length coordinate, it corresponds to time $t$ in the telegraph process. We decompose the probability $P[x(s)]$ of observing the projected conformation $x(s)$ as

[TABLE]

The functional $P_{\rm ideal}[x(s)]$ is the probability of observing the conformation $x(s)$ in an ensemble of ideal confined polymers, disregarding self avoidance. The functional $\mathscr{A}[x(s)]$ captures the effect of self avoidance. It equals the fraction of three-dimensional polymer configurations corresponding to $x(s)$ that contain no segments that collide with any other polymer segment.

When $D<\ell_{\rm P}$, the local conformation statistics are determined by Odijk’s theory for narrow channels Odijk (1983), while the global statistics are dominated by a random sequence of direction changes (hairpins) Odijk (2006). Neglecting self-avoidance, the separation between neighboring hairpin-bends is exponentially distributed Odijk (2006). On length scales much larger than the deflection length Odijk (1983) $\lambda\equiv(\ell_{\rm P}D^{2})^{1/3}$ the central-limit theorem assures that local alignment fluctuations are negligible Odijk (2008). These two facts imply that the ideal problem maps onto the one-dimensional telegraph process in Fig. 2. The correlation function of $v_{x}(s)$, the channel-axis component of the tangent vector of the ideal polymer decays exponentially SM

[TABLE]

The telegraph velocity has similar correlations:

[TABLE]

Comparing Eqs. (2) and (3) we see that the contour parameter $s$ maps to the time $t$ in the telegraph process, whereupon the polymer-contour length $L$ maps to the total time $T$ in the telegraph model. The parameter $a$ quantifies the tendency of the tangent vectors to align with the channel Werner et al. (2012). The parameter $g$ is the global persistence length Odijk (2006), characterizing the typical distance between hairpin turns. These parameters map to those of the telegraph process as follows: $a=v_{0}$ and $g=(2r)^{-1}$. We measured how $a$ and $g$ depend on the physical parameters of the full three-dimensional problem from simulations of confined ideal polymers. It turns out that it suffices to determine just two curves (Fig. 3a and b), since $a$ and $g/\ell_{\rm P}$ depend on $D/\ell_{\rm P}$ only.

Now consider the effect of self avoidance. In general it is very difficult to derive an expression for $\mathscr{A}[x(s)]$. But for a weakly self-avoiding polymer, the problem simplifies considerably when the channel is so narrow that interactions between the polymer and the channel wall cause the three-dimensional configurations to lose correlations. We show in the Supplemental Material SM that

[TABLE]

if $w\ll\ell_{\rm P}$. Here ${\mathscr{L}}(x)\,{\rm d}x$ is the total amount of contour in the interval $[x,x+{\rm d}x]$ van der Hofstad et al. (2003). The parameter ${\varepsilon}$ penalizes overlaps. It is determined by the probability that two short polymer segments overlapping in one dimension collide in three dimensions SM :

[TABLE]

The average is over the conformations of the confined ideal polymer, and $y$ and $z$ are the transverse channel coordinates of a short polymer segment of length $\ell$. Primed coordinates belong to a second, independent segment, and $v_{\rm ex}$ is the excluded volume. The excluded volume depends on the segment orientation. If $\ell\gg w$, we have $v_{\rm ex}=2w\ell^{2}\sin\theta$, where $\theta$ is the angle between the two segments Onsager (1949). Fig. 3c shows how ${\varepsilon}$ depends on $D/\ell_{\rm P}$, obtained by evaluating the average in (5) from three-dimensional simulations of confined ideal polymers SM . A single curve is sufficient to determine how ${\varepsilon}$ depends on the physical parameters, because ${\varepsilon}D^{2}/w$ is a function of $D/\ell_{\rm P}$ only (Supplemental Material SM ).

In the telegraph model self-avoidance is incorporated in the same way. Here, $\mathscr{L}$ has units of time/position. Eq. (4) then shows that ${\varepsilon}$ has units of position/time2. Since $r$ has units of time*-1*, and $v_{0}$ of position/time, the only dimensionless combination of ${\varepsilon}$, $r$, and $v_{0}$ is

[TABLE]

In the limit of large $T$ only $\alpha$ can have physical significance. Invoking our mapping between telegraph model and polymer problem we conclude that $\alpha$ is given by

[TABLE]

This parameter measures the expected number of overlaps between the two strands of a hairpin of length $g$.

Eq. (7) has two important consequences. First, Eq. (7) allows us to generalize the mapping to all channel widths up to $\ell_{\rm P}^{2}/w$. To show this, consider first the ideal part. Strictly speaking, the simple picture outlined above breaks down when $D\sim\ell_{\rm P}$ because the typical hairpin length $g$ becomes of the same order as $\ell_{\rm P}$. But consider how $\alpha$ changes as $D$ approaches $\ell_{\rm P}$ from below. For $w\ll\ell_{\rm P}$, the parameter $\alpha$ decreases below unity before $g=\ell_{\rm P}$ is reached, and for small $\alpha$ the precise nature of the local conformations is irrelevant. All that matters is that the ideal part is a diffusion process with exponentially decaying correlations of $v_{x}(t)$. Similarly, the local probability of collision is still $\tfrac{{\varepsilon}}{2}\,{\mathscr{L}}^{2}(x){\rm d}x$, because each segment pair collides independently. The latter assumption eventually breaks down at $D\approx\ell_{\rm P}^{2}/w$ since the transversal segment coordinates become correlated. But up to this point Eq. (4) is valid, as is Eq. (5).

Second, observables that are dimensionless in the telegraph model can only depend on $\alpha$, Eq. (7), in the limit of large $L$. This combination, $\alpha$, is plotted in Fig. 3d. It turns out that $\alpha D^{2}/(\ell_{\rm P}w)$ depends only on $D/\ell_{\rm P}$ SM . Now consider the average extension, $X$, and the variance about that average, $\sigma^{2}$. In the telegraph model these quantities have units of position and position2, and for large values of $L$ they must be proportional to $L$. We therefore conclude that the data must collapse as

[TABLE]

Here $f_{X}$ and $f_{\sigma}$ are universal scaling functions that depend only on $\alpha$.

We can numerically compute the form of these functions simply by simulating the telegraph model SM .

We have compared our theory to direct numerical simulations (DNS) of three-dimensional, confined, self-avoiding wormlike chains Tree et al. (2013a) using the PERM algorithm Grassberger (1997); Prellberg and Krawczyk (2004). Figs. 4a,b show that our theory quantitatively captures the DNS results for all values of $\ell_{\rm P}/w$ tested SM , up to $w/\ell_{\rm P}=0.4$. This agreement is remarkable, as the theory assumes weak self avoidance, $w\ll\ell_{\rm P}$.

Figs. 4c and d show the comparison between the results of the experiments summarized in Fig. 1 and our theory. The theory not only collapses the experimental data, but provides good quantitative agreement, in particular with the most recent experiments Gupta et al. (2015); Reinhart et al. (2015); Iarko et al. (2015). There is some scatter of the experimental data around the theoretical curve, but this is expected because the nanofluidic experiments are quite difficult to control.

In the limit $\alpha\ll 1$ our theory allows to map the problem to an uncorrelated weakly self-avoiding diffusion process van der Hofstad (1998); van der Hofstad et al. (2003). This follows from the fact that the correlation function in the telegraph model, Eq. (3), decays to zero before the next collision occurs, for $\alpha\ll 1$. As a result the ideal random walk is simply diffusive, with diffusion constant $\mathscr{D}=v_{0}^{2}/(2r)$. This has two consequences.

First, for $\alpha\ll 1$ observables depend on $v_{0}$ and $r$ only through the combination $\mathscr{D}$. Since the extension is linear in $L$ for large $L$, we deduce that $X/L$ can only depend on ${\varepsilon}$ and $\mathscr{D}$ in this limit. Since $\mathscr{D}$ has units of position2/time while ${\varepsilon}$ has units of position/time2 in the telegraph model, we see that the only possible combination is $X/L\propto({\varepsilon}\mathscr{D})^{1/3}$. This gives $X/(La)=f_{X}(\alpha)\propto\alpha^{1/3}$, explaining the power law in Fig. 4a and c. For the variance we conclude that $\sigma^{2}/L\propto\mathscr{D}$, independent of ${\varepsilon}$ (Fig. 4b and d). Alternatively we can deduce these scalings by a mean-field argument, indicating that fluctuations of $\mathscr{L}$ are negligible when $\alpha\ll 1$. Assuming that $\mathscr{L}\sim T/X$ we find for the extension distribution $P(X)\sim\exp[-rX^{2}/(2v_{0}^{2}T)-\tfrac{{\varepsilon}}{2}T^{2}/X]$, whereupon ${\rm d}\log P/{\rm d}X=0$ yields $X/(v_{0}T)\sim\alpha^{1/3}$. For the variance we obtain $\sigma^{2}r/(v_{0}^{2}T)\sim\alpha^{0}$.

Second, we can use the exact mathematical results derived in Ref. van der Hofstad (1998) to deduce the prefactors:

[TABLE]

as $\alpha\to 0$. The constraints for $c_{X}$ and $c_{\sigma}$ are rigorously proven mathematical bounds van der Hofstad (1998).

Now consider the limit of large $\alpha$. The extension $X$ tends to $La$ in this limit Odijk (2008), since the frequency of hairpins tends to zero. The variance decays as $\sigma^{2}\sim\alpha^{-3}$, as Fig. 4 shows. To deduce this power law, we estimate the variance of the strongly extended polymer as (number of hairpins) $\times$ (hairpin extension)2. To determine the number of hairpins, note that the expected number of collisions for a hairpin of contour length $h$ is $\sim\alpha\,h/g$. The resulting hairpin is therefore likely to survive the collision check only if $h$ is of the order $g/\alpha$ or smaller. This requires a second switch of direction within the length $g/\alpha$. This occurs with probability $(g/\alpha)/g=\alpha^{-1}$, so that the number of hairpins is $(L/g)\alpha^{-1}$. To obtain the hairpin extension we multiply its contour length $h\sim g/\alpha$ by its alignment $a$, so that the typical hairpin extension becomes $\sim ga/\alpha$. Therefore $\sigma^{2}\propto(L/g)\alpha^{-1}\times(ga/\alpha)^{2}=Lga^{2}\alpha^{-3}$ for large $\alpha$.

For very large values of $\alpha$, the theory fails [Fig. 4(b)] because hairpins are so rare that alignment fluctuations (not included in the telegraph model) dominate the variance Odijk (2008). This correction is taken into account simply by adding the variance in the extreme Odijk limit Burkhardt et al. (2010),

[TABLE]

Here $d_{\sigma}$ is a universal constant. By fitting the solid line in Fig. 4(b) for $\alpha>10$ we find $d_{\sigma}{\approx 0.09}$. We observe excellent agreement between this refined theory and the simulation data for $\ell_{\rm P}/w$ = 12. For the stiffer polymers ($\ell_{\rm P}/w$ = 36) still longer contour lengths are required to reach the large-$L$ limit and to reduce the statistical error.

Finally we show that our theory contains scaling laws derived earlier as particular asymptotic limits. In very narrow channels, $a\approx 1$ and $\langle\sin\theta\rangle\approx(D/\ell_{\rm P})^{1/3}$. Using these approximations in Eqs. (5) and (6) gives

[TABLE]

where $C\approx 1.95$ is a constant SM . The parameter $\xi$ appears in Odijk’s scaling theory Odijk (2008) and the extension scales as $X\sim\xi^{1/3}$ Odijk (2008) in this special limit. In wide channels, for $\ell_{\rm P}\ll D\ll\ell_{\rm P}^{2}/w$, we have that $a=1/\sqrt{3}$ and $g\approx\ell_{\rm P}$. Using diffusion approximations for the distribution of the polymer in the channel SM gives

[TABLE]

This is the result of Ref. Werner and Mehlig (2014), implying the same scaling $X\sim D^{-2/3}$ that Odijk’s scaling arguments Odijk (2008) predicts in this asymptotic limit. At first glance it is perhaps surprising that the two distinct limits (11) and (12) are described by the same random-walk process. After all, the three-dimensional polymer conformations are entirely different in the two regimes, described by invoking deflection segments, hairpins, and blobs. Our universal theory, by contrast, rests on the fact that the macroscopic statistics of a weakly interacting random walk do not depend on the microscopic details of the process Grosberg and Khokhlov (1994).

We can also conclude that the DNA experiments shown in Fig. 1 cannot obey the scalings $X\sim D^{-2/3}$ or $X\sim\xi^{1/3}$ because the experiments do not satisfy the strong inequalities $D\gg\ell_{\rm P}$ or $D\ll\ell_{\rm P}$ (see Fig. S-5 in the Supplemental Material SM ), and therefore do not reach the asymptotic limits required for these power laws to emerge. Our theory shows, and Fig. 4(a) confirms that $X\sim\alpha^{1/3}$ for small values of $\alpha$. But the parameter $\alpha$ depends upon $D$ and $\ell_{\rm P}$ in an intricate way via Eq. (7), in general not in a power-law fashion.

In summary, we have shown that DNA confinement in nanochannels can be modeled by a telegraph process, collapsing all of the data in terms of a scaling variable $\alpha$. Our theory brings to the fore universal properties of confined polymers in a good solvent in a way that is obscured by the prevailing scaling theories Daoud and De Gennes (1977); Odijk (1983, 2008); Werner and Mehlig (2014); Muralidhar et al. (2014); Dai et al. (2014). The distinction between deflection segments, hairpins, and blobs, which leads to the need to define separate regimes is not necessary. Rather, the statistics of the confined polymer chain for $D\lesssim\ell_{\rm P}^{2}/w$ adopt a universal behavior at sufficiently long length scales, independent of the details of the microscopic model.