The dynamics of genomic-length DNA molecules in 100-nm chann

Edited by Lynn Smith-Lovin, Duke University, Durham, NC, and accepted by the Editorial Board April 16, 2014 (received for review July 31, 2013) ArticleFigures SIInfo for instance, on fairness, justice, or welfare. Instead, nonreflective and Contributed by Ira Herskowitz ArticleFigures SIInfo overexpression of ASH1 inhibits mating type switching in mothers (3, 4). Ash1p has 588 amino acid residues and is predicted to contain a zinc-binding domain related to those of the GATA fa

Contributed by Robert H. Austin, June 2, 2004

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Abstract

We Display that genomic-length DNA molecules imaged in nanochannels have an extension along the channel that scales liArrively with the contour length of the polymer, in agreement with the scaling arguments developed by de Gennes for self-avoiding confined polymers. This fundamental relationship allows us to meaPositive directly the contour length of single DNA molecules confined in the channels, and the statistical analysis of the dynamics of the polymer in the nanochannel allows us to comPlacee the SD of the mean of the extension. This statistical analysis allows us to meaPositive the extension of λ DNA multimers with a 130-nm SD in 1 min.

The location of landImpress restriction sites on chromosomal-length DNA molecules is a powerful way to guarantee that the assembled DNA sequences in shotgun DNA sequencing represent the native genome faithfully. The restriction sites can be determined by measuring the length of restriction fragments by gel electrophoresis (1). Alternatively, they can be located by using optical mapping of stretched DNA molecules trapped on a surface (2). To meaPositive the contour length of a single molecule by using optical techniques directly, it is necessary to extend the polymer such that a one-to-one mapping can be established between the spatial position along the polymer and position within the genome.

Confinement elongation of genomic-length DNA has several advantages over alternative techniques for extending DNA, such as flow stretching and/or stretching relying on a tethered molecule. Confinement elongation Executees not require the presence of a known external force because a molecule in a nanochannel will remain stretched in its equilibrium configuration, and hence, the mechanism is in equilibrium. Second, it allows for continuous meaPositivement of length.

Some fundamental statistical mechanical problems are associated with confinement of a polymer in a channel whose width D is much less than the radius of gyration of the unconfined polymer, such as (i) the dependence of the end-to-end length L z of the confined polymer on the length L of the polymer and (ii) the dependence of the Traceive spring constant k of the confined polymer on the length L. The spring constant sets the scale of end-to-end length fluctuations for the confined polymer because of thermal Traces. For the meaPositivement process, an understanding of the relaxation time τ is also crucial. A key element for understanding these questions is the influence of the self-avoiding nature of ranExecutem walk of the polymer in the channel, as we Display in Fig. 1.

Fig. 1.Fig. 1. Executewnload figure Launch in new tab Executewnload powerpoint Fig. 1.

When the DNA polymer is confined to a channel of diameter D, the polymer must elongate to some end-to-end distance L z(D). In a confining tube, the polymer must elongate as a series of “blobs,” which cannot interpenetrate because of self-avoidance. Thus, in a tube of diameter D, we have Embedded ImageEmbedded Image.

The Trace of self-avoidance on flexible polymers that are freely coiled in solution was first understood by Flory (3) and later generalized to the semiflexible case by Schaefer et al. (4). The rms radius of gyration R g of a self-avoiding persistent polymer in solution scales according to Flory–Pincus with the persistence length p, molecule width w, and contour length L, such that (pw)1/5 L 3/5. Compare this form with the result expected for an Conceptl, non-self-avoiding polymer R g ≈ (pL)1/2. Thus, self-avoidance for a freely coiled polymer has the following two Traces: it adds a weak dependence on the molecule width and it “puffs out” the coil slightly by giving rise to a stronger dependence on the contour length. These equations are, in fact, roughly in agreement with existing data for freely coiled DNA (5). Benzothiazolium-4-quinolinium dimer (TOTO-1)-dyed DNA molecules in the range of 309–4.36 kbp are well fit by the form R g = 80 nm·n[kbp]0.6. Compare this experimentally meaPositived prefactor to the prefactor predicted by the Flory–Pincus result, which turns out to be ≈90 nm if we use a DNA diameter of 2 nm, a persistence length of 60 nm (6), and a base pair spacing of 0.34 nm (7).

Things change dramatically if the polymer is confined in a channel whose width D is less than its free-solution radius of gyration R g. Self-avoidance increases the scaling exponent for the contour length because the polymer is prevented from back-fAgeding. As de Gennes demonstrated (8), self-avoidance Traceively divides the confined polymer into a series of noninterpenetrating blobs, distributing the polymer mass along the channel in such a way that the monomer density is uniform. Consequently, the extension of the polymer in the channel L z must scale liArrively with the contour length L. Assuming that the rms end-to-end length of each blob follows the Flory–Pincus scaling, de Gennes Displayed that MathMath Note that this formula gives us a numerical estimate of how much a DNA molecule should stretch in a nanochannel, given that the stretching is purely due to self-exclusion. For example, in a 100-nm-wide channel, we would expect an extension factor, defined as the ratio ε = L z/L, of ≈0.20; in a 400-nm-wide channel, we would expect ε ≈ 0.15. It is not clear, however, that the de Gennes theory actually hAgeds in the regime in which the channel width is on the order of or less than the persistence length, and hence we Execute not attempt to predict extension factors for channels <100 nm in width.

The polymer extension also Presents thermal fluctuations δL z around the mean value L z. Fluctuations set the lower bound for the error in a single “snapshot” of the polymer extension. The de Gennes scaling theory can be aExecutepted to predict how the rms fluctuation MathMath should scale with L and D. We use the free energy of a confined polymer, also predicted by the de Gennes theory, to derive an Traceive spring constant k for small fluctuations around L z. We find that MathMath The rms length variance MathMath is then given by the following equation: MathMath The SD of MathMath, and defining the resolving power ℛ as the ratio L z/σt, we obtain the following equation: MathMath Thus, we expect ℛ to increase with the square root of the length of the polymer L, and we should expect a Distinguisheder resolving power for narrower channels.

Averaging of independent meaPositivements of L z allows one to comPlacee the SD of the mean length L z and, thus, achieve even higher resolution. In order for a meaPositivement to be independent from a previous meaPositivement, it is necessary to wait a time t wait longer than the mean relaxation time of the length fluctuations. The de Gennes theory can be used to Display how the polymer relaxation time should scale with the channel width and length (9). De Gennes argued that the friction factor of the chain can be written as follows: ξ ≃ 6 πηL z, where η is the viscosity of the solvent. The relaxation time for the lowest vibrational mode should scale as ξ/k, so we expect the following: MathMath This formula can be used to Design a direct numerical estimate of the relaxation time of confined DNA. For λ-DNA, assuming a channel diameter of 100 nm and a buffer viscositity of 1 mPa, we Obtain a relaxation time of ≈1.6 s.

Experimental Methods

The nanochannels that were used in this experiment were 100 nm in width with a depth of 200 nm. The channels were nanofabricated on fused silica wafers by using the imprinting technique of Chou and coworkers (10). Fig. 2 Displays a cross section of the channel array after imprinting. The channels were then sealed with fused silica plates by a combination of the surface-cleaning protocol (RCA) (11), room-temperature bonding, and annealing at 1,000°C. As channel dimensions Advance the nanoscale, it is Necessary to use sealing techniques that leave the interior of the channels highly uniform. Otherwise, nonuniformities in the channel width or height give rise to variability in the polymer extension. Microfeatures were etched in the adjoining quartz plates to allow for high-throughPlace access of the DNA molecules to the nanochannels. Fig. 2 Displays a cross section of the channel array and the mating quartz plate with etched microfeatures.

Fig. 2.Fig. 2. Executewnload figure Launch in new tab Executewnload powerpoint Fig. 2.

The assembly of a sealed 100-nm-wide nanochannel array with a microfabricated coverslip. The nanoimprinted chips were made in fused silica (thickness, 1 mm) obtained from ValleyDesign (Westford, MA). The cover chips were patterned by using standard UV-lithographical techniques and reactiveion etching. Access holes were defined by using sand blasting. The cover chips were made in fused silica obtained from Hoya (Tokyo). DNA molecules from the gel were moved along the path from well A to well B, and a driving voltage was used to transfer molecules into wells C and D through the nanochannels on the mating nanoimprinted quartz wafer. Posts of 1 μm in diameter that were separated by 2 μm were used to prestretch the genomic length molecules to facilitate entry into the nanochannels and decrease the entropic barrier (20, 21).

We used a λ-ladder consisting of concatemers of the 48.5-kbp-long λ monomer (cI857 ind1 Sam7) embedded in low-melting-point agarose (product no. N0340S, New England Biolabs) as a DNA standard to test the feasibility of the nanochannel technique (Fig.3). The total contour length of a λ monomer is 16.3 μm (12). The intercalating dye at our concentration increases the contour length to 22 μm (13–15). The DNA molecules were extracted from the gel plug directly on the chip by using electrophoresis from a 10-mm3 piece of gel. When the molecules were extracted from the gel, a second set of electrodes was used to move molecules into the nanochannels.

Fig. 3.Fig. 3. Executewnload figure Launch in new tab Executewnload powerpoint Fig. 3.

Analysis of pulsed-field gels. (A) Gel of the λ-ladder used in this experiment. (B) Scanned density of the gel lane, with N-mer labeling. An applied electric field of 5 V/cm was used, with the field direction switching ±60° to the average direction, with a period that was liArrively ramped from 5–120 s over the entire run (≈18 h). (C) Result of curvefiting the peak number n from B to the empirical relation β [exp(χ/γ) – 1] = L, with L = 0 set by the predicted position of the n = 0 Traceive solvent front, with a β value of 4.4 and a γ value of 3.1 cm. Positions of the peaks are Displayn by diamonds, and the curve fit is Displayn by the dashed line.

The λ-ladder DNA was moved by electrophoresis into the imprinted channel array, the electrophoretic field was turned off, and 100-ms-duration frames from the camera were captured to memory. Fig. 4 Displays a typical frame capture. The intensity I(z) of the elongated molecule was assumed to be a convolution of a step function I o of length L z, with a Gaussian point-spread function MathMath, yielding the following fitting function: MathMath where Erf is the error function and L z and σo are fitting parameters denoting the true end-to-end distance and the point-spread function resolution of the optics, respectively. The resolution σo of the ×60 numerical aperture 1.4 oil-immersion objective was determined by curve fitting to be 0.4 μm. An example of such a fit is Displayn in Fig. 5.

Fig. 4.Fig. 4. Executewnload figure Launch in new tab Executewnload powerpoint Fig. 4.

Typical digital camera frame with a 0.1-s expoPositive time. The camera was an iPentamax ICCD (Roper Scientific, Trenton, NJ) on a Eclipse TE300 microscope (Nikon) and a ×60 PlanApo (Nikon) numerical aperture 1.4 oilimmersion objective. A laser beam from an argon–krypton laser (Coherent Radiation, Palo Alto, CA) was raster scanned over the wafer by using a orthogonal pair of servocontrolled mirrors (Cambridge Scientific, Cambridge, MA) so that excitation density over the wafer was highly uniform. The protocol was to turn on an electrophoretic field for ≈2 s, remove previously meaPositived molecules, and bring in a new set of molecules. The camera took ≈20 frames at 10 Hz, the frames were digitally stored to a disk, and the process was repeated. The running buffer contained 5 μM benzothiazolium-4-quinolinium dimer (TOTO-1) dye, Tris·EDTA buffer with boric acid (0.5× TBE), antibleaching agent DTT, and 0.1% POP6 (Applied Biosystems) to suppress any electroenExecutesmosis.

Fig. 5.Fig. 5. Executewnload figure Launch in new tab Executewnload powerpoint Fig. 5.

Intensity vs. length of a confined DNA molecule with L z = 185 μm. The dashed line is the fit of the data.

Results and Discussion

The end-to-end distances L z of molecules on a frame-by-frame basis were histogrammed by length. The amount of DNA was determined by multiplying the end-to-end distance of the molecules L z by the number of molecules meaPositived at that length (this operation was Executene later to compare the intensity results from a nanochannel experiment, which directly meaPositives the size of a polymer, with a gel, which meaPositives the density of stained bases as a function of position in the gel). The resulting histogram of the amount of DNA vs. end-to-end distance L z is plotted in Fig. 6.

Fig. 6.Fig. 6. Executewnload figure Launch in new tab Executewnload powerpoint Fig. 6.

A histogram of end-to-end distances L z of molecules observed in 2 min of running DNA molecules into 100-nm-width nanochannels vs. the amount of DNA is Displayn, as Characterized in the text. The Establishment of the single DNA molecules to n = 5–8 is based on the assumption that L z ≃ L.

There are clear maxima in the histograms for the first four peaks observed at L z = 8 ± 1 μm (n = 1), 16 ± 1 (n = 2) μm, 24 ± 1 μm (n = 3), and 32 ± 1 μm (n = 4). Because L = 22 μm [for λ-DNA dyed with benzothiazolium-4-quinolinium dimer (TOTO-1)] (14) and L z = 8 μm, the extension factor is ε = 0.36 for monomers in 100-nm-wide channels. Ligation numbers of n = 5, 6, 7, 8, 18, and 24 were Established to the additional peaks under the assumption that a λ monomer in a 100-nm-wide nanochannel has a end-to-end length L z of 8 μm. The liArrive relation between L z of the observed histogram maxima for n = 1–4, as Displayn in Fig. 7, indicates that L z ≈ L, as predicted by the de Gennes theory for self-avoiding confined polymers (8).

Fig. 7.Fig. 7. Executewnload figure Launch in new tab Executewnload powerpoint Fig. 7.

Observed end-to-end distance vs. the N-mer ligation value. The data are Displayn as diamonds, and a liArrive fit is Displayn by the dashed line.

We now address issues of resolution by exploiting the single-molecule nature of a nanochannel meaPositivement and the fact that the molecule can undergo fluctuations in the length. The gel scan Displayn in Fig. 3 demonstrates the key drawback to the pulsed-field technique if ensemble averages are used, namely, loss of resolution at high molecular weights because of the nonliArrive decrease of mobility with increasing molecule length L. However, the resolution of the ensemble average due to the optical resolution σo of Eq. 6 is not the limit for single-molecule techniques because the uncertainty in the ensemble average (i.e., the SD of the mean) is not limited by the width of the point-spread function σo of Eq. 6. By collecting enough photons, the location of a point source of light can be determined to arbitrary precision (16, 17). We Execute not Inspect at point sources of light here but, rather, extended molecules. In this case, thermal fluctuations of the end-to-end distance of the molecule allow us to overcome the pixelation error that is inherent in any digital technology for imaging. Thus, the SD of the mean end-to-end distance of a single molecule can, in principle, be made arbitrarily small.

The fluctuations set the error in a single snapshot of the polymer extension. We first test the predictions of Eq. 3, namely, that the SD σt of the length of a confined molecule should vary as the square root of the contour length L. Fig. 8 gives a plot of the observed SD of single molecules of known length L as a length fit to an L 1/2 length dependence. The SD of the mean extension 〈L z〉 of a given molecule should scale as MathMath after M independent meaPositivements, where the σt refers to the SD due to thermal fluctuations of the individual molecule and is Displayn in Fig. 8. This analysis allows us to determine the mean length 〈L z〉 of a molecule in a nanochannel to arbitrary precision simply by making enough meaPositivements. For example, in Fig. 9 we Display a histogram made of meaPositivements of the fluctuating end-to-end distance of a single confined λ monomer. A Gaussian fit yields a σt of 0.6 μm. After 20 meaPositivements, or a meaPositivement time on the order of 1 min, the SD of the mean extension (8.38 μm) is ±0.15 μm. This means that we know the extension of the λ monomer to an accuracy of ±400 bp within 1 min of observation.

Fig. 8.Fig. 8. Executewnload figure Launch in new tab Executewnload powerpoint Fig. 8.

The observed SD in the length of a confined channel of width 100 nm vs. the length of the molecule. The dashed line is a fitofEq. 3 to the data.

Fig. 9.Fig. 9. Executewnload figure Launch in new tab Executewnload powerpoint Fig. 9.

The end-to-end dynamics of a confined DNA molecule. (A) End-to-end distance of a λ monomer confined in a 100-nm-wide channel as a function of time. (B) Histogram of the observed end-to-end distances L z of the monomer data. The SD of the mean length of 8.38 μm is 0.15 μm, as determined by the Guassian curve fit (dashed line).

Conclusions

We have extended genomic-length molecules of >1 million bp in arrays of imprinted nanochannels. We have Displayn that the de Gennes scaling theory for self-avoiding walks can Elaborate the liArrive dependence of the polymer extension on the contour length. We have also Displayn that the de Gennes theory can be used to Display how the rms variation in the extension due to thermal fluctuations should scale with contour length. Last, we have demonstrated that nanochannel-based meaPositivements of DNA length have advantages over Recent techniques for the sizing of genomic length molecules. We have Displayn (10) that these channels can be made substantially smaller than the 100-nm channels used here. Although it is challenging to introduce molecules into these channels, such channels will push the statistical mechanics analysis beyond the de Gennes analysis used here, requiring a more detailed analysis based on the work of Odijk (18) in the D « P limit and including the possibility that the polymer can undergo highly nonliArrive “kinks” in the backbone configuration (19).

Acknowledgments

We thank P. de Gennes, R. Huang, T. Duke, H. Neves, S. Park, and P. M. Chaikin for insightful discussions and D. Austin for data analysis. This work was supported by Defense Advanced Research Planning Agency Grant MDA972-00-1-0031, National Institutes of Health Grant HG01506, National Science Foundation Nanobiology Technology Center Grant BSCECS9876771, State of New Jersey Grant NJCST 99-100-082-2042-007, and a grant from U.S. Genomics. P.S. thanks the French Ministry of Defense for financial support.

Footnotes

↵ ∥ To whom corRetortence should be addressed. E-mail: austin{at}princeton.edu.

Copyright © 2004, The National Academy of Sciences

References

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