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Dec 4, 2015 - entropic spring force to one that is confinement dependent. In a strongly confining slit with a depth equal to one persistence length we...
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Stretching DNA Molecules in Strongly Confining Nanofluidic Slits Alexander C. Klepinger, Madeline K. Greenier, and Stephen L. Levy* Department of Physics, Binghamton University, Binghamton, New York 13902, United States S Supporting Information *

ABSTRACT: We experimentally investigate the stretching and relaxation of individual double-stranded DNA molecules in nanofluidic slits with depths that span the regime between moderate and strong confinement. DNA molecules are stretched by the application of a quadrupole-like homogeneous elongational electric field. In a moderately confining slit we verify the previously observed existence of two distinct relaxation times resulting from the transition from a bulk-like entropic spring force to one that is confinement dependent. In a strongly confining slit with a depth equal to one persistence length we observe the return to a coil−stretch transition that is governed by a single strain rate related to the confined spring constant. By measuring the equilibrium extension as a function of the applied strain rate as well as the relaxation dynamics from a highly stretched initial state, we are able to infer a hydrodynamic friction coefficient in the moderately and strongly confining slits that is in good agreement for a long DNA molecule. Our results are helpful for informing theoretical models of the force−extension relation for semiflexible polymers in quasi-two-dimensional space.

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INTRODUCTION Microfabricated devices allow for the precise manipulation of biopolymers using geometrical confinement and well-defined flow fields. These devices have proven useful for investigating the dynamics and conformation of individual DNA molecules using fluorescent microscopy1 and for technological applications like separating and barcoding DNA molecules.2−4 Confinement alters both properties in a nontrivial manner partly due to the increased importance of self-interactions relative to a bulk environment5,6 and due to the screening of hydrodynamic interactions by the confining walls.7,8 There has been a substantial increase in our understanding of the fundamental processes involved in the deformation and stretching of polymers over the past decade,9,10 but subtle issues related to confinement remain unresolved. Fluorescent microscopy has shed light on a longstanding difficulty in describing the extension of an unconfined polymer in an elongational flow,11−13 where a velocity or electric field gradient above a critical value can quickly stretch the polymer to a large fraction of its contour length. The predicted coil− stretch transition14 occurs when the stretching force is greater than the linear region of the polymer’s entropic elastic force that resists the deformation. The entropic force required to reach a given extension for a semiflexible polymer has been shown to follow the Kratky−Porod, or worm-like chain, force law,15 and experiment agrees well16 with an approximate expression17 for this law. Questions remain concerning the coil−stretch transition and applicability of the worm-like chain law, which does not include excluded volume interactions, in strongly confining structures. We refer to a nanoslit as a confining fluidic structure with a © XXXX American Chemical Society

height, typically between tens and hundreds of nanometers, which is much less than its width. Though widely used in interpreting experimental results, the force−extension relationship derived from the Kratky−Porod model has limited validity for self-avoiding semiflexible polymers in two dimensions.18,19 The importance of determining the extension under a small applied force for a self-avoiding worm-like chain in free solution is also recognized.20−22 Bakajin et al. showed that the relaxation time of a highly stretched DNA molecule in a nanoslit increased as the height decreased due to hydrodynamic friction between the polymer and the walls.23 The importance of the hydrodynamic interaction was confirmed using a Brownian dynamics simulation24 and by theory.25 Several authors have proposed modified versions of the force−extension relation for polymers as they transition from weak to strong confinement in a slit.26−28 We will relate the theory relevant for explaining the steady-state extension under an applied force and the relaxation time of a stretched semiflexible polymer in a confining slit. Both require understanding how the entropic spring constant and hydrodynamic friction coefficient of a molecule are altered in confinement. In a series of experiments and simulations, the Doyle group has shown that the coil−stretch transition of double-stranded DNA has an interesting dependence on the depth of the confining nanoslit.29−32 Using a homogeneous planar elongational electric field, they have demonstrated that it requires less force to stretch a DNA molecule to a given fractional extension Received: August 1, 2015 Revised: November 19, 2015

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DOI: 10.1021/acs.macromol.5b01712 Macromolecules XXXX, XXX, XXX−XXX

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these regimes is predicted to occur when the relaxing tensile blobs become equal in size to the height of the slit.29 At that point relaxation proceeds via the random diffusion of the selfexcluding blobs. The crossover occurs when the self-avoiding blobs are aligned, leading to a fractional crossover extension of

in a slit than in bulk. A DNA molecule in a slit with a height less than the molecule’s radius of gyration in bulk stretches under less applied force. This occurs because the molecule’s entropic spring constant is reduced in confinement due to enhanced excluded volume interactions and because the molecule’s hydrodynamic friction coefficient is increased. Together these lead to the existence of two relaxation times in moderately confining slits, within the linear force−extension regime, as will be explained in more detail. The electrophoretic stretching force exerted by the extensional electric field is proportional to this friction coefficient under the assumption of hydrodynamic equivalence.33 This assumption has been shown to be slightly inaccurate for highly stretched molecules, however.34 Several aspects of these results can be explained using the blob model of de Gennes.35 This model has been used to quantitatively describe the conformation and relaxation of DNA molecules near equilibrium in nanoslits.29,36−39 In this model, the elastic spring constant kslit for small deformations scales as 1/Rg2, where Rg is the radius of gyration in the slit. In the blob model, Rg is determined by a self-avoiding random walk of blobs in two dimensions, where the blob size is proportional to the slit height h. For a semiflexible polymer with persistence length lp and width w, the contour length in a self-avoiding blob Lblob is given by Lblob ≈ h5/3(wlp)−1/3. We use the approximately equal to sign to indicate that possible numeric factors are ignored. This leads to the following scaling relation37 between kslit and h kslit ≈ h1/2L−3/2(wlp)−1/2

Xc /L ≈ (wlp)1/3 h−2/3

A Brownian dynamics simulation verified the existence of the two relaxation times and their dependence on excluded volume interactions in moderate confinement.32 The independence of the fractional crossover extension on molecular weight has also been demonstrated.29 More recently, it has been proposed that there is an extended de Gennes regime for slits with h < lp2/w (and h > lp), where the traditional blobs do not contain enough persistence lengths to be self-avoiding.42 In this regime, blobs are ellipsoidal with height h and radius B given by B ≈ lp(h/w)1/2. The contour length in an ellipsoidal blob is Lell ≈ hlp/w, and the crossover extension occurs at Xec /L ≈ (w/H)1/2. Interestingly, the equilibrium size (and spring constant) are the same in the classic and extended de Gennes regimes. However, the friction in the extended de Gennes regime should scale as ζEdG ≈ ηB(L /Lell ) ≈ ηL(w/h)1/2

(4)

which agrees with the measured exponent for the scaling of the height from stretching and diffusion experiments.29 It is important to note that both w and lp depend on the ionic strength of the buffer (see Supporting Information). Theory and experiment have shown that a polymer stretches significantly when the Deborah number De = ε̇τ exceeds a critical value of 0.5, where ε̇ is the strain rate.43 The Deborah number is also sometimes referred to as the Weissenberg number in elongational flow.10 The coil−stretch transition occurs when the applied stretching force is larger than the entropic spring force. In moderate confinement, the coil− stretch transition is more gradual than in bulk due to a softened spring constant. When stretching a polymer in confinement, the transition from the softer spring constant to the bulk spring constant previously described leads to a double-peaked structure in the fluctuation of the extension as a function of applied strain rate31 due to a slowdown of the polymer dynamics.44,45 The effects of strong confinement (h ≤ lp) on the coil− stretch transition remain to be investigated. Based on a balance of the conformational energy of the entropic spring and stretching energy, it has been predicted that a single peak in the fluctuations will reappear in strong confinement.31 This should occur when the transition from the softened spring constant to the bulk value occurs beyond the linear regime of the entropic spring. For a relatively high ionic strength buffer, this transition occurs at about a depth of one persistence length. The predicted peak in the fluctuations should be observed at a critical value of Dec = kslit/2kbulk, where the bulk spring constant kbulk for an ideal polymer is 3kT/2Llp. We note that an additional transition from the extended de Gennes regime to the Odijk regime46 is predicted to occur when h < lp though its experimental observation in slit-like confinement has not been demonstrated reproducibly.38,39 We investigate the effects of strong confinement on the coil− stretch transition in this report. We manipulate individual DNA molecules in a homogeneous planar extensional field in crosschannel slits of moderately and strongly confining heights,

(1)

where L is the contour length. Also, intermolecular hydrodynamic interactions are reduced in a slit, acting only over a length scale given by the slit height.40 The screened interactions result in an increased hydrodynamic friction coefficient inside a slit ζdG that also enhances the stretch of a molecule under a given force relative to its stretch in bulk. The friction coefficient is given by37 ζdG ≈ ηh(L /L blob) ≈ ηL(wlp)1/3 h−2/3

(3)

(2)

38

The measured scaling of the diffusion (inverse friction) with slit height has an exponent closer to 0.5. This has been explained41 within the blob theory framework by modifying the form of the pair correlation function for distances less than lp. The Doyle group has also demonstrated the existence of two distinct longest relaxation times, referred to as τI at higher extension and τII near equilibrium, for a DNA molecule in a slit.29−31 Both relaxation times are important when stretching molecules at fractional extensions less than 30%. According to the worm-like chain model, the force to extend a DNA molecule in bulk is linear with the extension up to about 30%. It was shown that the higher extension relaxation time τI governs the dynamics, still within this linear regime, when the DNA molecule is stretched to a large enough extent so that it is not sterically interacting with the slit walls. The longer relaxation time τII governs relaxation near equilibrium and agrees fairly well with blob theory predictions in slits,38 given by τII ∼ ζdG/ kslit. They have estimated based on theory and a comparison to experimental results that τI ∼ L2.2h−1/2. The existence of multiple relaxation times stems from the existence of two spring constants: the bulk constant that applies when a molecule is stretched such that its tensile blobs are smaller than the height of the slit and the confined spring constant that applies near equilibrium. The crossover between B

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Figure 1. (a) Schematic illustration of the trapping region and coordinate system. The east and west reservoirs are connected to variable potentiometers to control the applied strain rate. (b) Optical micrograph of the trapping region in the 165 nm deep device. (c) The extension of a T4 molecule in the 165 and 50 nm deep slits as a function of time with the east and west reservoirs held at 27 V relative to the north and south reservoirs and (d) the relaxation of the same molecule from its steady-state extension in each slit as a function of time, with zero applied voltage. The width of the entire image region (black rectangle) for (c) and (d) is 56 μm. measured in several regions using a contact profilometer (Veeco, Dektak) with a quoted resolution of less than a nanometer. The depths of the devices described were estimated to be 165 and 53 nm with an uncertainty of 5% due to variation in the etch rate across the wafer (we use 50 nm when referring to the latter device for simplicity). Access holes to the fluidic structures were created using an abrasive sandblaster. Wafers were cleaned in NanoStrip (Cyantek) overnight and in an RCA bath prior to being touch bonded to a 170 um thick fused silica cover wafer (Mark Optics) and placed in a furnace at 1100 C for several hours.48 Pipette tips were affixed over the access holes with a silicone adhesive (Dow Corning 732), and devices were initially filled with a mixture of ethanol and deionized water. Devices were filled with a tris-borate-EDTA (5X-TBE) buffer consisting of 445 mM tris base, 445 mM boric acid, and 10 mM EDTA (Sigma-Aldrich) with 4 vol % β-mercaptoethanol (BME). The ionic strength of this buffer is estimated to be 160 mM based on known dissociation constants for the acid and base components (including the BME), and its pH was measured to be 8.2 (Symphony, VWR). The buffer was pressure driven through a filter with 20 nm diameter pore size (Whatman) before use. The dynamic viscosity of this buffer was inferred from its measured kinematic viscosity (Cannon-Manning Semi-Micro viscometer) to be 1.39 mPa s at 20.2 C (approximately 40% larger than water at this temperature). We note that the dynamic viscosity of 5X-TBE without the added BME was measured to be 30% larger than water. We applied 25 V (BK Precision) across the device for 30 min before running experiments to stabilize the ionic current. Bacteriophage T4 DNA (166 kbp, Wako Chemicals) was stained with YOYO-1 dye (Invitrogen) at a ratio of 1 dye molecule for every 5 base pairs in the 5X-TBE buffer at a final DNA concentration of 30 pM. Devices were loaded onto an inverted microscope (IX71, Olympus) with a light source (Exfo), suitable fluorescent filters (Semrock, 3035), and a 60 magnification, 1.4 numerical aperture oil-immersion objective (Olympus). Images were recorded with an exposure time of 150 ms on a camera (Photometrics, Evolve 512) connected to a PC using custom software (Labview). To run experiments several microliters of the T4 DNA solution was added to the east reservoir of a given device (the height of the solution in each of the four reservoirs was equalized). A positive voltage was

which are approximately three and one persistence lengths, respectively. The slits are fabricated in fused silica wafers by photolithography and thin film etching. Figure 1a shows a schematic representation of the setup, an optical micrograph of the central region of a stretching device (Figure 1b), and images of the extension and relaxation of a fluorescently labeled DNA molecule as a function of time in devices with 165 and 50 nm slit heights, respectively (Figure 1c,d). We stretch fluorescently stained T4 DNA molecules (165.6 kbp) that have an estimated contour length of 73 μm, after staining with YOYO-1 dye at a ratio of 1 dye molecule for every 5 base pairs,47 in a relatively high ionic strength buffer (see Experimental Methods). A DNA molecule is stretched by applying a positive potential to the east and west reservoirs relative to the north and south reservoirs that are held at ground. Variable resistors connected as shown allow a molecule to be trapped at the stagnation point and stretched to a large fraction of its contour length with minimal feedback control. A molecule is shown in Figure 1c as it is stretched at a similar strain rate ε̇ in the 165 and 50 nm slit. After the molecule reaches a steady-state extension under this strain rate, it is shown relaxing as a function of time in Figure 1d in each slit depth. Both the steady-state extension and relaxation dynamics depend on the depth of the device as expected.

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EXPERIMENTAL METHODS

Devices for stretching DNA were fabricated on 100 mm diameter, 500 μm thick fused silica wafers (Mark Optics). The fabrication consisted of a single layer of contact photolithography followed by reactive ion etching with a CHF3/O2 plasma. Devices consist of perpendicular intersecting fluidic structures that are each approximately 1.8 cm in length and 40 μm in width outside of the stretching region where the channels intersect. The hyperbolic curve defining the geometry at the intersection region is defined by y = ab/x, where the parameters a and b are 50 and 40 μm, respectively. The depths of the structures were C

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Macromolecules applied to the west reservoir with the east held at ground to drive DNA through the device. We note that a much larger voltage was required to drive DNA into the 50 nm device than the 165 nm device. We are not aware of the reason for this. A large fraction of the T4 molecules were observed to be fragmented when they passed the central trapping region, as others have observed with this length DNA.49 A small number of molecules were discarded during the subsequent analysis if they were observed to have equilibrium extensions that were more than five standard deviations from the average. When an unfragmented molecule was observed in the trapping region, we centered it at the stagnation point using variable potentiometers as shown in Figure 1a. We waited approximately 30 s before we applied an extensional electric field and recorded images. The extensional electric field was applied so that molecules were stretched along the east−west direction. This ensured that DNA molecules in the reservoir or connecting channel were not driven through the trapping region during the stretching process. In some cases, molecules were again stretched to a steady-state extension at a large value of ε̇ after they had been used for data collection to ensure that they were still unfragmented. The images were analyzed using custom software written in Matlab (Mathworks). Each image was convolved with a Gaussian and boxcar filter to enhance the signal to background. Then connected pixels above a threshold determined on a per image basis were identified as done previously.50 These pixels defined the identified DNA molecule, and the variables defined in the following section were determined using the raw (unconvolved) pixel values in each frame. Based on the geometry of the fluidic channels, it is expected that the velocity v of a point charge varies linearly with position as vj = μEj = ε̇j, where μ is the electrophoretic mobility, E is the electric field, and j denotes the position along the x or y dimension. We verified that the electric field increases linearly with position in both the x and y directions by measuring the trajectory of fluorescently stained lambda DNA (48.5 kbp) molecules, or fragments, at different applied voltages (see Supporting Information) as they accelerate through the trapping region. We observed a linear relation between the strain rate and applied voltage over the relevant experimental range.

Figure 2. (a) Relaxation of a stretched DNA molecule as a function of time in the 50 nm (red square) and 165 nm (blue circle) deep nanoslit. The solid line represents a fit to a simple dumbbell model of the relaxation process that is used to infer the hydrodynamic friction coefficient at each depth. (b) The fractional squared extension, corrected for the equilibrium extension, in the 50 nm (red square) and 165 nm (blue circle) deep slits. The solid lines at small and large times indicate the regions where the relaxation times (τI and τII) were respectively determined in each slit.

nm slit, where it is equal to 46% or 30% for the de Gennes (eq 3) or extended de Gennes regimes, respectively. Again we have ignored a prefactor of order unity in each estimate. This makes it difficult to define a priori fitting regions for the relaxation time. Consequently, the regions were determined based on these estimates and visual inspection of the data in Figure 2b. The fitting regions and resulting values for the relaxation times are shown in Table 1. Zoomed-in plots for the fits of each

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RESULTS AND DISCUSSION In order to calculate the low and high extension Deborah numbers for a given strain rate, we measured the respective relaxation time constants at each slit height. We applied a strain rate that resulted in a large fractional extension (greater than 50%) and measured the extension Xex as a function of time for approximately 10−15 molecules in each device as they relaxed to equilibrium over 80 s. The extension Xex is measured as the length of the molecule projected onto the x-axis and is normalized by the contour length L (73 μm). We plot the average normalized extension at each time for each device depth separately in Figure 2a. To extract the relaxation times, we plot (Xex2 − Xeq2)/L2 versus time on a semilog scale, where Xeq is the equilibrium extension measured over the last 5% of the observation time, as shown in Figure 2b. We fit the logarithm of (Xex2 − Xeq2)/L2 to a first-order polynomial10 in order to extract the relevant relaxation time τi using linear leastsquares regression, where i is I or II. The free parameters in the fitting algorithm are the offset and slope (τi), and the results of the fit are shown as black lines in Figure 2b. As mentioned, there is some uncertainty concerning the value of the extension at the crossover between τI and τII depending on whether the de Gennes or extended de Gennes regime applies. The boundary of h ≈ lp2/w occurs at approximately 560 nm based on the parameters of the DNA at the 160 mM ionic strength buffer used (see Supporting Information for details). The difference in the expected fractional crossover extension becomes significant in the 50

Table 1. Measured Relaxation Times for T4 DNA height (nm)

τi

fit region (ti, tf) (s)

165 165 50 50

τI τII τI τII

(3.0, 5.8) (20, 60) (10, 15) (30, 60)

fit region (Xiex, Xfex)/L (0.35, (0.12, (0.37, (0.23,

0.27) 0.07) 0.32) 0.15)

time (s) 5.0 21.2 12.6 11.5

relaxation time are shown in the Supporting Information. Qualitatively, the polynomial fits agree with the data reasonably well over the sometimes narrow range of the fit. The average value of Xeq is 6% and 14% for the 165 and 50 nm deep device, respectively. This difference is larger than expected based on the scaling of equilibrium size with height in the de Gennes regime.37 Using eq 1, we would expect the extension in the 50 nm slit to be about 35% larger than in the 165 nm slit. However, Xeq in the 50 nm slit is significantly larger than the measured value of Xex at a strain rate of zero (the first data point in Figure 3a), indicating that a longer observation time is required before equilibrium is reached in the highly confining slit. We typically ended observation of the relaxation process after 90 s to avoid artifacts from photobleaching. D

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Figure 3. (a) Fractional extension as a function of strain rate for the 50 nm (red squares) and 165 nm (blue circle) deep nanoslits. The color scheme for the nanoslits is the same for all panels of the figure. (b) Fractional extension, (c) fluctuation in the fractional extension, and (d) root-mean-square angle of the principal axis of the radius of gyration as a function of DeI = ε̇τI in each of the slits. The dashed line in (b) indicates the expected extension as a function of DeI once the hydrodynamic friction coefficient has been inferred by balancing the electrophoretic stretching force with a generalized entropic spring force at each slit depth.

Balducci et al.29 have estimated that τI follows a scaling of based on a comparison of their results to those of Bakajin h et al.23 This also agrees with the measured value of the scaling of diffusion (inverse friction) with height at equilibrium in slits, which is slightly different from the de Gennes blob value as discussed. Balducci et al. assume that the entropic spring constant is given by the bulk value and that it has no dependence on the slit height. We observe a stronger scaling of τI with height (τI ∼ h−0.8) when comparing our results in the 165 and 50 nm deep slits. However, the scaling we observe agrees well with that measured for τI by both Tang et al.31 and Balducci et al.30 in moderate confinement for T4 DNA in a device with similar geometry. We note that the measured relaxation times at high and low extension in the 50 nm slit are very similar. However, it is clear from Figure 2b that a single fit covering both regions at once would not be reasonable given the nonlinear (on the log scale) intermediate region. It is, however, odd that we observe a smaller value for τII in the strongly confining slit relative to the moderately confining one. We discuss one potential reason that may explain this fact shortly. A second potential reason is that the slit height is in the transition region between the extended de Gennes and Odijk regimes.28,42,51 However, we have not observed a transition from the expected blob relaxation time near equilibrium at this depth previously.38 Additional comparisons to previous results after accounting for changes in the ionic concentration and viscosity of the buffer can be found in the Supporting Information. We use the values for the relaxation time of τI in Table 1 to define the higher extension Deborah number, DeI = ε̇τI. We have ignored any effects from the relaxation of polarized counterions or due to electrode polarization during

this process since their expected time scale is orders of magnitude smaller than those measured.52 Additionally, we estimate the friction coefficient of a relaxing T4 molecule using a simple dumbbell model of two beads separated by a spring. We assume that the entropic force Fs at a fractional extension of x = Xex/L is given by the following expression from Chen et al.26 for the force−extension relation of a wormlike chain in slit-like confinement

−1/2

Fls kT

⎤ ⎡ 1 1 = ⎢(x − x0) + (1 − x)−2 − (1 − x0)−2 ⎥ ⎦ ⎣ 4 4

(5)

where l∥ is the longitudinal correlation length and x0 is the equilibrium fractional extension. There are few examples of force−extension relations in the literature that allow for a nonzero extension at zero force. Using Brownian dynamics simulations of highly confined chains, Chen et al. found a phenomenological relationship for l∥/lp as a function of lp/h. For the 165 and 50 nm deep slits, the value of l∥/lp is approximately 1.1 and 1.5, respectively. By assuming that the entropic force on a bead is balanced by the hydrodynamic drag force, we can solve for the change in fractional extension δx = x − (2Fs/ζb)δt as a function of time, where ζb is the friction coefficient of a bead. We numerically minimize the squared difference between the measured and expected fractional extension as a function of time to determine the optimum value of ζb and assume that the total friction is 2ζb. We use eq 5 for Fs and begin the iteration from the measured extension at time zero. Interestingly, we note that the scaling of the extension with force follows roughly Xex ∼ (Fslp/kT)1/2 for eq 5 over the intermediate force range where Fslp/kT ∼ 0.5, for an assumed fractional equilibrium extension of 10% (see E

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where the steady-state extension was measured, we determine a value of ζb that minimizes the squared difference between the expected and measured extension. This leads to values of the friction coefficient that are similar to those previously measured (each within 10%) and then allows us to compute the extension for any value of ε̇ as shown in the solid lines of Figure 3b. We can equivalently compare the value of Fe obtained for the measured steady-state extension Xex to the force−extension relation as shown in the Supporting Information. In either case, the extension at a given Deborah number is consistent with the shape of Fs(Xex) using the inferred conformation independent hydrodynamic friction coefficient. It is significant to note that a critical strain rate governs the coil−stretch transition in bulk, and it has been shown that this transition softens (occurs over a broader range of DeI values) in confinement.31 However, we observe that in the strongly confining slit the transition has become sharper (occurs over a smaller range of DeI values), though it is still less sharp than in bulk. This is evident if one compares the slope of the dashed lines near DeI ∼ 0.5 in Figure 3b. We plot the average fluctuation in the fractional extension σ(Xex)/L as a function of DeI for each slit depth in Figure 3c. We define σ(Xex) as the standard deviation of the extension sampled for all molecules at a given ε̇ in a given slit as described previously. In practice, we found that computing Xex and σ(Xex) from the sampled data (compared to the data sampled at the acquired frames per second) did not result in a noticeable change to these figures. We also computed σ(Xex) for each molecule individually at a given value of ε̇ in a given slit height. The uncertainty in σ(Xex) shown in Figure 3c is computed as the standard error of the mean of these values, where the average is over the molecules for the prescribed data point. We observe two peaks in the plot of σ(Xex)/L for the 165 nm deep slit as has been observed by the Doyle group30,31 in similar confinement. The first peak, at approximately DeI = 0.10, is evident but not very prominent. The existence of two peaks is related to the existence of two linear entropic spring constants, a softer confined spring constant and a bulk spring constant, for extensions less than 30% as described earlier. However, the double-peaked structure is much less pronounced in the highly confining slit as expected. It was anticipated that as the confining dimension is decreased that a single peak in the fluctuation of the extension should again be observed near the lower value of the critical DeI (near 0.10). This is due to the balancing of the electrophoretic stretching force with the linear region of the confined spring force. This balancing should result in a sharper coil−stretch transition at the lower critical value of DeI. This is also somewhat evident in Figure 3b as discussed. However, the extent of the linear regime of the spring force at low extension in strong confinement is open to debate.19 Also, the increased equilibrium size of the molecule as the slit height decreases may blur the transition region between the regimes (see Supporting Information). Consequently, we interpret the results of Figure 3b,c as being consistent with the expected behavior predicted for the transition to strong confinement, though some of the quantitative details remain unresolved. In Figure 3d, we plot the average molecular orientation in the two slits as a function of DeI. Specifically, we plot the average of the root-mean-square value that the angle ϕ of the principal axis of the in-plane radius of gyration38 makes with the x-axis. This is determined for all molecules at a given value of ε̇ in a given slit height and sampled with the same frequency as previously mentioned. The angle is constrained to be within the

Supporting Information). This differs from the Kratky−Porod force−extension relation in 2 or 3 dimensions where the extension is linear with the force until the extension exceeds 30%. Hsu and Binder have found argued that Xex ∼ (Fslp/kT)1/3 for a semiflexible polymer in 2 dimensions over a similar intermediate force range.19 In their simulations, the transition from the linear regime to this Pincus regime occurs at much smaller values of the extension than where the nonlinear transition occurs in the Kratky−Porod model.19 It is possible that the smaller value of τII observed in the strongly confining slit is due to this qualitative difference in the lower extension region of the entropic force in quasi-two-dimensions. We find values for the total friction of 60 and 110 fN s/μm for the 165 and 50 nm slit, respectively. The predicted extension as a function of time for each depth is shown as a solid curve in Figure 2a using the inferred value for the relevant hydrodynamic friction coefficient. The combination of the slit force−extension relation and friction coefficient describe the time-dependent extension well. We assume that the friction coefficient is conformation independent throughout the relaxation process in a given height slit since the hydrodynamic interactions are screened and the molecule is free-draining. This is reasonable but approximate. Bakajin et al. estimated the friction coefficient per length in strong confinement23 as ξ′ = 2πη/ln(4h/πd) where they assumed that the DNA molecule diameter d is 2.5 nm and found that ξ′ = 1.4η described their data. In the 50 nm slit, we find a corresponding value of ξ′ = 1.1η. We can also compare the ratio of the coefficients in each slit height (1.83) to expectations of the scaling of friction with height from the blob theories. The scaling from eq 2 in classical blob theory yields a ratio of 2.13, from eq 4 in the extended blob regime a ratio of 1.76 and using a correction to the classical blob theory41 a ratio of 1.75. We plot the fractional extension as a function of strain rate in Figure 3a. At each strain rate we plot the average fractional extension of approximately 10−15 molecules after they have reached steady-state extension. We average the extension for a given molecule by sampling the extension every τI seconds after a Hencky strain (ε̇t) of either 5 or 10 units has been reached. For the smaller ε̇ values in the 50 nm slit we used the lower value to prevent photobleaching and maintain a similar signalto-noise ratio in identifying the DNA molecule. The average fractional extension shown in Figure 3a at a given value of ε̇ is the average of the samples for all molecules for the given slit height. The uncertainty for each value is calculated as the standard error of the mean. It is evident that a molecule extends to a significantly larger fractional extension as a function of strain rate in the strongly confining relative to moderately confining slit. We plot the average fractional extension as a function of DeI in Figure 3b since that variable has been shown to govern the coil−stretch transition in confinement.30 As a cross-check on the hydrodynamic friction coefficient measured from the relaxation process, we can also infer the coefficient by using the steady-state extension at a given DeI in conjunction with the confined force−extension relation. In the dumbbell model of a polymer with the beads separated by a distance Q in an extensional field, the electrophoretic stretching force53 is given by Fe = ζbε̇Q cos(2ϕ)/2, where ϕ is the angle of the dumbbell axis with respect to the x-axis. At steady-state extension, we approximate this as Fe ≈ ζbε̇Xex/2 and recognize that this force should be balanced by the force−extension relation of eq 5. By equating these expressions at each value of ε̇ F

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range −π/2 to π/2. We verify that the average value of ϕ is zero as expected. Since the probability distribution is flat over this range, the root-mean-square of ϕ is expected to be 0.9 rad. In both slit heights the molecules begin to align with the x-axis at small values of DeI. It is evident that increasing the confinement causes orientation of the DNA molecule in the direction of the electric field at a lower value of DeI.

ACKNOWLEDGMENTS This work was performed in part at the Binghamton University Nanofabrication Facility and at the Cornell NanoScale Facility, a member of the National Nanotechnology Infrastructure Network which is supported by the National Science Foundation (Grant ECCS-0335765). This research was supported by NSF Grant DMR-1351283. We thank Melissa Stanke for her work in obtaining preliminary data by stretching DNA molecules in nanofluidic devices.

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CONCLUSIONS In conclusion, we measured the extension and relaxation of T4 DNA molecules in a moderately and strongly confining nanofluidic slit. Molecules were stretched by applying a homogeneous elongational electric field. The steady-state extension, fluctuations in the extension, and molecular orientation were determined as a function of the applied strain rate and Deborah number. Relaxation times and friction coefficients were measured in each slit that agreed with scaling laws and previous estimates. In moderate confinement we observed a broad coil−stretch transition with two critical Deborah numbers where the extension fluctuations were maxima due to the existence of a softer confined spring constant and a bulk spring constant for extensions less than 30%. In strong confinement, we observed a transition to a single critical Deborah number governing the coil−stretch relationship based on the confined spring constant. Our results are useful to compare with the predicted force−extension relation for semiflexible polymers in quasi-two-dimensions. It would be interesting to probe the coil−stretch transition and this relation in stronger confinement at the onset of the expected Odijk regime. Our results are relevant for devising nanofluidic devices to manipulate DNA molecules in strong confinement where many biologically interesting phenomena occur.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.5b01712. Description and figures of the measurement of the strain rate at a given voltage; zoomed-in regions of the squared fractional extension as a function of time where the slopes are fit to obtain the relaxation times; a comparison to previous results of the steady-state extension and fluctuation in the extension in moderate confinement as a function of Deborah number; a comparison of the relaxation times to previous results corrected for buffer ionic strength and viscosity differences; a plot of the measured force−extension relation and alternative models of the force extension relation for semiflexible polymers (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected] (S.L.L.). Present Address

A.C.K.: Department of Physics, The Ohio State University, Columbus, OH 43210. Notes

The authors declare no competing financial interest. G

DOI: 10.1021/acs.macromol.5b01712 Macromolecules XXXX, XXX, XXX−XXX

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DOI: 10.1021/acs.macromol.5b01712 Macromolecules XXXX, XXX, XXX−XXX