Transition between Two Regimes Describing Internal Fluctuation of DNA in a Nanochannel

We measure the thermal fluctuation of the internal segments of a piece of DNA confined in a nanochannel about 50100 nm wide. This local thermodynamic property is key to accurate measurement of distances in genomic analysis. For DNA in 100 nm channels, we observe a critical length scale 10 m for the mean extension of internal segments, below which the de Gennes' theory describes the fluctuations with no fitting parameters, and above which the fluctuation data falls into Odijk's deflection theory regime. By analyzing the probability distributions of the extensions of the internal segments, we infer that folded structures of length 150250 nm, separated by 10 m exist in the confined DNA during the transition between the two regimes. For 50 nm channels we find that the fluctuation is significantly reduced since the Odijk regime appears earlier. This is critical for genomic analysis. We further propose a more detailed theory based on small fluctuations and incorporating the effects of confinement to explicitly calculate the statistical properties of the internal fluctuations. Our theory is applicable to polymers with heterogeneous mechanical properties confined in non-uniform channels. We show that existing theories for the end-to-end extension/fluctuation of polymers can be used to study the internal fluctuations only when the contour length of the polymer is many times larger than its persistence length. Finally, our results suggest that introducing nicks in the DNA will not change its fluctuation behavior when the nick density is below 1 nick per kbp DNA.


Introduction
Stretching DNA in nanochannels has emerged as an important technique for separating DNA, performing genome mapping, and also studying repressor-DNA interactions, etc [1][2][3]. On the other hand, DNA confined in nanochannels also serves as a simplified model for studying single polymer behavior in concentrated polymeric solutions and melts [4,5]. For these reasons, mechanical behaviors of DNA inside nanochannels have attracted a longstanding interest. The two most well-known scaling theories in this field are those described by de Gennes [5] and by Odijk [6]. de Gennes' blob theory, which was later generalized by Schaefer and Pincus [7], assumes that the channel width D is much greater than the persistence length j p of the polymer. It models the moderately confined DNA as a chain of spherical blobs inside a cylindrical channel and gives the following expression for the end-to-end extension SxT of the polymer [5,7,8]: where L, w are the contour length and effective molecule width of the DNA respectively. The prefactor A is found to be close to unity [9]. Odijk's theory, on the other hand, works for DNA under strong confinement in which D%j p . In this regime, the polymer is deflected back and forth by the channel walls and the end-to-end extension is predicted to be [6]: where a 0~0 :17 is a constant whose value was determined recently by simulations [10]. Aside from the scaling theories, Wang and Gao [11] showed that the end-to-end extension of a strongly confined polymer in the Odijk regime can be derived analytically by modeling the confinement effect as a quadratic potential U~1=2Jjr r \ j 2 . Here J is the stiffness of the effective quadratic potential, which depends on the channel width D, andr r \ is the transverse displacement of the polymer from the axis of the nanochannel. Wang and Gao considered a confined chain under end-to-end applied force F and obtained an expression for the total extension SxT as a function of J and F . We set F~0 pN, substitute the relation between J and D (see Supporting Information) into their expression, and find: which is the same as Eq.2, confirming the scaling theory of Odijk, and at the same time validating the use of quadratic confinement potentials in the strongly confined regime. Both de Gennes' and Odijk's theories have been tested by experiments as well as simulations over the years [10,[12][13][14][15][16]. However, most of the studies so far have focused on the properties of the entire DNA, for example, the end-to-end extension SxT, the corresponding end-to-end fluctuation s x , and also the relaxation time t of the entire DNA etc. Local properties of a confined polymer, on the other hand, like the extension and fluctuation of its internal segments, are rarely investigated. In fact, local conformation and alignment of the confined DNA have been probed only recently [17,18]. It is also not well understood whether the existing theories developed for an entire piece of DNA can be applied locally for its internal segments. These are important issues because, if one considers the case of genome mapping, it is the local fluctuation of the internal segments that determines the resolution of the mapping.
In this paper, we measure the longitudinal internal fluctuation of a piece of DNA confined in rectangular channels about 50{100 nm wide. We show that neither de Gennes' blob theory nor Odijk's deflection theory can completely describe the measured internal fluctuation versus mean extension profile. A critical length scale of *10 mm for the mean extension is observed, below which the internal DNA segments are more 'blob'-like, and above which Odijk's deflection theory works better. From the histograms of extension of the internal segments, we further infer that there exist folded structures of length 150{250 nm separated by *10 mm along the backbone of the DNA during the transition between the two regimes. To justify the use of existing theories for studying the internal fluctuation, we focus on the Odijk regime and propose a method to explicitly calculate the internal fluctuation of a strongly confined DNA. We model the confinement effects by quadratic potentials and show that one can use the existing theories for end-to-end extension/ fluctuation to describe the internal segments of the DNA when the contour length of the polymer is many times larger than its persistence length. Our model, which views the confined DNA as a discrete wormlike chain, can describe the fluctuations of heterogeneous polymers confined in non-uniform channels. It is also capable of capturing effects, like the influence of nicking sites on the DNA fluctuation profiles, which we will discuss at the end of the paper.

Results and Discussion
To visualize the internal segments, dye-labeled (Alexa-546) nucleotides are introduced into the backbones of the nicked l DNA (48:5 kbp, L&16:5mm), T4 DNA (166 kbp, L&56:4mm) and bacterial artificial chromosome (BAC) human DNA clones (MCF7 BAC clone 9I10, fragmented, full length *180kbp, L&61:2mm) ( Fig. 1) [19]. The DNA molecules are then driven by electric field into the nanochannels. With the Alexa-546 labels excited by light, extension of each internal segment is recorded frame-by-frame. Average extension SxT and the root mean square (rms) fluctuation s~ffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Sx 2 T{SxT 2 p for each internal segmenet are calculated and plotted in the s{SxT profile.
In Fig. 2, we first show the result for l DNA confined in a 80 nm|130 nm channel. The maximum SxT, which is roughly the mean extension of the entire DNA, is about 10mm, in agreement with the measurements of Tegenfeldt et al [12]. The internal fluctuation s increases with SxT with a 0:5 power law. This 0:5 power law and even the magnitude of the fluctuation can be well captured by de Gennes' theory (discussed below) with no fitting parameters.
The longitudinal fluctuation of the confined DNA in de Gennes' theory can be evaluated using the effective stiffness k eff of the polymer: s 2~k B T=k eff %(4=15)L(wj p D) 1=3 [12,20]. Using this expression and Eq.1 to eliminate L, we get the relation between s and SxT: Therefore, de Gennes' theory predicts a 0:5 power law for the s{SxT profile. It is interesting to note that the prefactor in Eq.4 depends only on the channel width D, but not on the effective molecule width w, nor on the persistence length j p . This implies that the s{SxT profile is independent of the ionic strength of the experimental buffer. To compare the theory with the measured internal fluctuation, we plot Eq.4 together with the experimental data in Fig. 2. Surprisingly, the data matches with the theory very well without any fitting parameters. Both the 0:5 power law and the magnitude of the fluctuation are correctly predicted by Eq.4. de Gennes' theory also gives the distribution of the extension P(x), which we can compare to our measurement. We consider the recently proposed ''renormalized'' Flory-type free energy F for a confined polymer [21] and its corresponding prediction of the longitudinal fluctuation: where b~1=k B T, A, B are two constants, N, g are the total number of monomers and the number of monomers inside a blob respectively [21]. Both of the relations can be rewritten in terms of SxT (which is the solution of LF =Lx~0) as: with C~(2A) 2=3 B 1=3 being a constant. The probability distribution P(x) is therefore: Here P 0 is a constant determined by the normalization condition.
In our experiments, we record the extension x of each internal segment frame-by-frame and then calculate the distribution P(x) for each segment. Fig. 3 shows the measured P(x) for two internal segments and their fitting results to Eq.7 (red). The result again implies that, for l DNA confined in a 80 nm|130 nm channel, the behavior of the internal segments can be well captured by de Gennes' theory. Moreover, by fitting the distribution P(x) to Eq.7, we obtain the constant C, which, when plugged back into Eq.6-2, yields: s~0:58 02 nm). Therefore, starting from the ''renormalized'' Flory-type free energy Eq.5, we recover Eq.4 with the same prefactor. This indicates that the prefactor in Eq.4 is quite accurate although it is derived from a scaling theory. It also explains why Eq.4 matches with the measured s{SxT profile without any fitting parameters (Fig. 2). It is important to note that, for l DNA confined in a 80 nm|130 nm channel, the maximum SxT is less than *10 mm (Fig. 2). We shall show next that for longer DNA whose maximum SxT is greater than *10 mm, the measurement no longer agrees with de Gennes' theory. In particular, the 0.5 power law in the s{SxT profile is lost. Fig. 4A shows the s{SxT profile for the internal segments of T4 DNA in a 80 nm|130 nm channel. The maximum SxT, which is roughly the mean extension of the entire DNA, is about 30mm, in agreement with the simulation result of Jung et al [14]. Fitting of s*SxT c to the experimental data yields c~0:19, which is very different from the prediction of de Gennes' theory (Eq.4). Similar results are found for DNA in channels of different sizes: c~0:15 for T4 DNA confined in 60 nm|100 nm channels (Fig. 4B) and c~0:11 for l DNA in 50 nm|70 nm channels (Fig. 4C). In all these cases the maximum SxT is greater than 10mm. We note, however, that in Fig. 4, the experimental data for segments with SxT 10mm still matches with de Gennes' theory (except for the 50|70 nm channel case, which we will explain later). It is the data with SxT * > 10mm that deviates significantly from de Gennes' prediction. In fact, if we plot the fluctuation results for short segments with SxT 10mm for l and T4 DNA together, the two profiles are almost identical, satisfying de Gennes' theory (see Supporting Information Fig. S1).
To rule out the possibility that the observed difference between l DNA and T4 DNA stems from sequence variations, we perform the same experiments on the bacterial artificial chromosome (BAC) human DNA clones (MCF7 BAC clone 9I10), which also  has maximum SxT * > 10mm. As shown in Fig. 5, the results for the BAC DNA are almost identical to those for the T4 DNA. In particular, for small SxTv10mm, both match with de Gennes' prediction without any fitting parameters, while for SxTw10mm, both identically deviate from de Gennes' prediction. This suggests that the deviation from de Gennes' theory for long internal segments truly stems from segment size, not from sequence variations.
To better understand the deviation from de Gennes' prediction, we further look i nto the local structures of the confined DNA. Odijk showed recently that even in a 135 nm channel, DNA can fold back on itself, giving rise to a global persist ence length much larger than 50 nm, the intrinsic persistence length of the DNA [18,22]. Because of this, Odjik argued that the transition from Odijk's regime to de Gennes' regime could be delayed with the increase of the channel size [18]. To check whether such local folded structures exist in the DNA in our experiments, we measure the extension distribution P(x) for each single internal segment (see ''Materials and Methods'' for details). We find that for most internal segments whose mean extension is longer than 10mm, the distribution P(x) shows two or more peaks (Fig. 6B-C). From this observation, we infer that there indeed exist some folded structures in those internal segments -one peak in the distribution corresponds to the folded configuration, and the second peak corresponds to the extended configuration (Fig. 6). The existence of folded structures can be also inferred from the typical extension x versus time plot as shown in Fig. 6D, where the steps in x correspond to different states of the internal segments. Furthermore, we find that in the distribution P(x), the measured distances between any two peaks are always integral multiples of 400{500 nm, indicating that the difference in extension of a single folded structure and its extended form is about 500 nm, ten times the persistence length of the DNA. This further implies that each branch of the folded structure is about 150{250 nm, if we assume each folded structure has two (loop) or three (hairpin) branches (Fig. 6). Also, by checking the location of the internal segments that show multiple-peak distributions, we find that the folded structures are separated by *10 mm, which roughly agrees with the value of SxT above which de Gennes' theory fails to match with the experimental data (Fig. 4). In the following we show that for SxT * > 10mm the fluctuation data is better described by Odijk's deflection theory.
To exactly (rather than in a scaling sense) evaluate the fluctuation of DNA in the Odijk deflection regime, we extend the theory recently developed by Wang and Gao [11]. This theory represents the DNA as a strongly confined wormlike chain (fluctuating elastic rod) subjected to an additional end-to-end force F and produces the relation between the mean extension SxT and J, the stiffness of the effective confinement potential (which is a function of the channel width D): where again, k B T is the thermal energy, k is the bending modulus of the polymer, and in a rectangular channel the stiffness of the confinement potential can be expressed as J~4c 4 k B T= ½ (k 1=4 D 2 ) 4=3 , with c being a constant. Using Eq.8, we calculate the effective stiffness of the DNA as k eff~L SxT=LF ð Þ {1 , and then evaluate the fluctuation as s 2~k B T=k eff :  , the data SxT * v 10mm agrees with de Gennes's theory (red, no fitting parameters). Deviation from de Gennes' theory begins at a critical SxT*10mm, above which the data falls into the black curve predicted by the deflection theories of Odijk [6], Wang and Gao [11]. For tighter channels (C), the transition occurs earlier with most data falling in the deflection regime. doi:10.1371/journal.pone.0016890.g004 Leaving c as a free parameter, we fit Eq.9 to the experimental data with SxTw10mm in Fig. 4A-C (black curves) and obtain c~1:03, 0:94 and 0:99 respectively. For the BAC DNA confined in 80 nm|130 nm channels shown in Fig. 5, we obtain c~0:9 from a similar fit. The fact that all the four sets of experimental data for different channel widths yield the same c&1 makes sense because c is expected to be a universal constant independent of D.
Moreover, the constant c comes from the expression for the free energy of confined chains in the Odijk regime and it has been estimated by Burkhardt to be c~1:1 [23], which is very close to our fitting results. This strongly suggests that in the large mean extension regime SxTw10mm, the DNA segments are better described by the deflection theory. Furthermore, from Fig. 4A to C, we observe that the length of the error bars decreases with the decrease of the channel size. The reason for this may be that for moderately confined DNA, the local folded structures can form and unravel with comparable rates, as indicated by the similar height of the two peaks in the distribution in Fig. 6B-C. Therefore, the behaviors of the confined polymer is a competition between de Gennes' type and Odijk type regimes and the error bar is large. As the channel size becomes smaller, Odijk's theory begins to dominate, resulting in smaller error bars.
By integrating the force-extension relation Eq.8, we obtain the free energy expression G(x) in the Odijk (or Wang and Gao) deflection regime (see Supporting Information), which further leads to the distribution for the extension P(x): where A~L 2 =4j p , B~4c 2 j 1=3 p =D 4=3 and P 0 is the normalization factor. We fit this expression to the right peaks in Fig. 6B-C and find that reasonable parameters (L&15mm, j p &50 nm) give excellent matches with the measured probability distributions in experiments. In fact, we can use this free energy expression to understand the transition from a different point of view. We note that the internal segments are expected to stay in the regime with lower free energy, and that regime transition occurs when the free energies in the two regimes are equal. By comparing the free energies in the two regimes, we draw a phase diagram on the L{D plane in Fig. 7. The result shows that as D decreases, the transition length L decreases. Theoretically, the phase diagram involves an undetermined constant, which we fit such that transition occurs in the range L&8{12mm when D~100 nm. Then the result shows that at D~60 nm, the transition length is 3{5mm, which roughly agrees with our experimental result for l DNA in a 50 nm|70 nm channel (Fig. 4C). The phase diagram shows that transition from de Gennes' to Odijk's regime can occur when D decreases with L fixed, or when L increases with D fixed.
We also measure the end-to-end extension for DNA with different lengths (longer than 10 microns) in a 60 nm|100 nm channel and the result agrees with Odijk's theory (Fig. S3).
In the above analysis, we have applied the theories (de Gennes, Odijk, Wang and Gao) for the end-to-end extension/fluctuation to evaluate the internal, or local extension/fluctuation of a confined DNA. The assumption behind this is that when the internal segments are much longer than the persistence length of the DNA, the behavior of the segments is not very different from that of the entire DNA (with the same length) because the boundary conditions do not play a significant role [24][25][26]. To verify such an assumption, we explicitly calculate the internal fluctuation in Odijk's regime by extending a theory we developed earlier [26], and then compare our results to the theories developed for an entire piece of DNA.
Following the procedure in ref. [26], we model the polymer as a confined discrete N{segment wormlike chain, or fluctuating elastic rod (Fig. 8). The Hamiltonian consists of 3 terms (Eq.11): (1) bending energy, (2) confinement energy, and (3) potential energy of an end-to-end applied force as shown in Fig. 8.
In the bending energy term, k(s) is the bending modulus of the DNA and it can vary along the arc length s so that the polymer is not necessarily homogeneous in mechanical properties.t t is the tangent vector along the polymer. For the confinement potential term, we follow Wang and Gao's approach [11] and use an effective quadratic energy characterized by the coefficient J, with y being the transverse displacement. In general, J can be a function of the arc length s in case the confinement is not uniform. Also, for 3D chains in rectangular channels, J can be different in the two transverse directions. For the potential energy term, we consider the chain subjected to an end-to-end force F , which can be set to zero if no force is applied. Dx~x(L){x(0) is the end-toend extension of the chain. Up to a second order approximation, the Hamiltonian can be written in matrix form as shown in Eq.12, with h i being the discretized tangent angles and K being the N|N stiffness matrix of the chain [26]. It has been shown that when there are no constraints on twist (as is the case here), thermodynamic properties of a 3D chain can be easily generated from those of two 2D chains [26]. Therefore, for simplicity, here we describe the theory for 2D chains and plot the results for the corresponding 3D chains.
To get the internal fluctuation, we first need to calculate (1) the partition function, and (2) the angle fluctuation Sh i h j T. These are evaluated in the ''Materials and Methods'' section. Finally, for any internal segment between node i and node j of the discrete chain, the mean extension Sx ij T and the corresponding rms fluctuation can be explicitly calculated as: where l is the segment length of the discrete chain. In Fig. 9, we consider DNA in 60 nm|60 nm channels and plot s ij versus Sx ij T for all the pairs of internal nodes (i,j) and see if the profiles match with the theories developed for the entire piece of DNA. Fig. 9(A) shows the result for a chain with contour length L~10mm, which is much larger than its persistence length j p~5 0 nm. The internal fluctuation profile agrees exactly with Eq.9, which is derived for the end-to-end fluctuations. In particular, all the data collapses into a single curve with 0:5 power law. As the contour length of the polymer decreases, however, (Fig. 9B-D), the internal fluctuation profile begins to scatter around the curve for the end-to-end fluctuation. This implies that, for short chains, the magnitude of internal fluctuation can be different, even if two internal segments have the same mean extension. The magnitude of the fluctuation depends strongly on where the internal segment is located. In fact, we show in Fig. 10 that the internal segments located at the two boundaries have larger fluctuation because they have more freedom to fluctuate compared to the segments inside the chain. The strong boundary effects on short chains (such as, DNA with contour length 0.6-7 mm) have been discussed by several groups recently [24][25][26].
Our results suggest that the accuracy of DNA sizing depends on  the DNA contour length. For a short DNA with contour length Lv1mm confined in a 60 nm|60 nm channel, the uncertainty of the measurement will be high. For the experimental results we discussed earlier, the l DNA, T4 DNA and BAC DNA all have contour lengths of tens of microns, for which boundary effects can be neglected. Therefore, it is safe to use the formulae for end-toend extension/fluctuation to estimate the internal properties of the confined DNA in our experiments.
To measure the internal fluctuation, we have introduced nicks into the DNA so that internal sites along the DNA can be labeled. Since the theory discussed above allows for arbitrary bending modulus k(s) as a function of the arc length s, we can model the effect of nicking by setting k~0 on some nodes of the discrete chain and see whether the nicks have significant effects on the behavior of the DNA. For simplicity, we assume here that the nicks are equally spaced along the chain. Fig. 11 shows that the fluctuation profile does not significantly deviate from the homogeneous chain with uniform k when there are less than 50 nicks along a 18mm chain (*50 kbp DNA in a 60 nm|60 nm channel). In our experiments, the fluorescent tagging is introduced at the nicking endonuclease recognition sequence sites, which have much lower density than 1 nick/kbp in l, T4 and BAC DNA. Therefore, the nicks will not significantly affect the DNA internal fluctuation.
To summarize, in this paper, we have investigated the thermal fluctuations of the internal segments of a piece of confined DNA in a nanochannel. The channel size is on the order of the persistence length of the DNA and we have compared the fluctuation data to several theories in literature. We have found that for channel widths on the order of 100nm there exists a critical length scale *10 mm for the mean extension of an internal segment below which the de Gennes' theory describes the internal fluctuations and above which the data agree better with Odijk's deflection theory. For long DNAs confined in nanochannels we have inferred that there are folded structures whose branches are about 3 times the persistence length of DNA which are separated by segments with mean extension *10mm. We surmise that these folded structures are indicative of a  A and B), data from internal segments of various locations of the chain collapse on the a curve with 0:5 power law (light green). The result agrees with Eq.9 (blue), which is derived for the endto-end fluctuation of a confined DNA. For short DNA however (C and D), no power law is found as data from various locations of the chain do not collapse onto a single curve (light green). Therefore, formulae derived for the end-to-end fluctuation of the confined DNA, such as Eq.9 (blue), cannot be used for internal fluctuation. The boundary effect is so significant that the rms fluctuation s not only depends on SxT, but also on the location of the internal segments. doi:10.1371/journal.pone.0016890.g009  transition from the Odijk regime, in which the DNA is relatively straight, to the deGennes regime, in which the DNA is more blob-like. We have also presented a more detailed theory based on small fluctuations and incorporating the effects of confinement. We have shown that one can use the existing theories for end-to-end extension/fluctuations to study the statistical properties of internal segments only when the contour length of the chain is much larger than the persistence length of the molecule so that boundary effects play no role. Our calculations suggest that introducing nicks into the DNA can change its fluctuation behavior if the density of nicks is greater than about 1 nick per kbp DNA.

Materials and Methods
Sequence specific labeling and DNA staining

Loading DNA into nanochannels
Fabrication of silicon based nanochannel chips has been described elsewhere [27,28]. The DNA sample is diluted by 2 times using the flow buffer consisting of 1|TBE, 3.6% Tween, and 10% Polyvinylpyrrolidone (PVP). Ultrapure distilled water is used for making solutions (Invitrogen Corp., Ultrapure water). The DNA molecules are driven by electric field (3{5 V) at the port of entrance of the chip and allowed to populate there for 2{3 minutes [29]. Under higher voltage (*10 V), the populated molecules are moved to the locos and then through the micro pillar structure of the chip to convert from a compact globular conformation to an open relaxed one. At the 300nm channel area the molecules adopt a more relaxed linear form with some heterogeneity on the backbone. With one end entering the nanochannel under the electric field, the DNA molecules elongate to a linear conformation with almost homogeneous backbone. Most of the structural heterogeneity progressively disappears as it interacted with the nanochannels, adopting fully confined equilibrium conformation after the field is off (relaxation time 10{15 s). A buffer consisting 0.5|TBE, 1.8% Tween 20, 5% PVP has been used to flow the DNA molecules resulting in a stretch of 65%.

Recording and calculations
The intensity profile I(x,y) of each Alexa-546 label is fitted by a 2D Gaussian function to determine the position of the label (x c ,y c ) in the channel (Fig. 1B). The position of each internal label is followed frame-by-frame at a time interval of about 160 ms. The probability distribution, the mean value and the corresponding standard deviation of the distance between each pair of internal labels are calculated.

Partition function and angle fluctuation
The partition function for a confined DNA, whose Hamiltonian is expressed in Eq.12, is: where N is the number of segments in the discrete chain. The angle fluctuation or correlation is the Boltzmann weighted average of (h i h j ) over all the configurations [26,30]: Using Eq.15, we can explicitly calculate the mean extension and fluctuation of the internal segments (Eq.13-14). Figure S1 s versus SxT profile for the SxTƒ10mm region.  Figure S3 Mean end-to-end extension SxT versus contour length L of confined DNA in a 60 nm|100 nm channel. The fitting result is x~0:5L, which is consistent with the prediction of the Odijk deflection theory: x~0:7L.

Supporting Information
Text S1 (PDF)