Correlated Fluctuations of DNA between Nanofluidic Entropic Traps

Jun 19, 2015 - Macromolecules , 2015, 48 (13), pp 4742–4747 ... of the channel, complex nanofluidic environments contain regions of varying dimensio...
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Correlated Fluctuations of DNA between Nanofluidic Entropic Traps Alexander R. Klotz,*,† Mikhail Mamaev,† Lyndon Duong,† Hendrick W. de Haan,‡ and Walter W. Reisner† †

Department of Physics, McGill University, Montreal, QC H3A 0G4, Canada Faculty of Science, University of Ontario Institute of Technology, Oshawa, ON L1H 7K4, Canada

‡

ABSTRACT: Here we explore thermally driven contour fluctuations within a single DNA chain partitioned between two embedded cavity reservoirs in a nanofluidic slit. Analysis of integrated cavity intensity suggests that contour is exchanged dynamically between the reservoirs via modes that resemble the symmetric and antisymmetric modes of a coupled harmonic oscillator. The relaxation time of the modes is measured as a function of cavity width and spacing. Langevin dynamics simulations reproduce our observations and motivate a free energy model with a blob-type hydrodynamic friction, developed and used to deduce how the measured relaxation times for the modes depend on device parameters. The relaxation time of the antisymmetric mode was found to be consistent with an excluded volume-based stiffness while the faster symmetric mode depends on additional entropic and elastic parameters.

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INTRODUCTION The study of the static properties of DNA in model nanofluidic geometries such as nanoslits and nanochannels has revealed how static polymer conformation depends on device dimension.1 In contrast to nanoslit and nanochannel geometries that have only a single degree of freedom, i.e., the vertical height of the slit or width of the channel, complex nanofluidic environments contain regions of varying dimension and confinement scale. These multiple degrees of freedom can be tuned in order to control the static and dynamic molecular properties of single molecules.2 In addition, complex environments lead to new properties such as single chain trapping at locally open sites in the structure and partitioning of contour between multiple trapping sites. Natural sieving media such as gels are examples of complex nanofluidic environments as are micro/nanofabricated entropic trapping arrays3 and the zeromode waveguide structures4 used in some next-generation sequencing technologies. Of particular interest are the time scales of internal fluctuations of contour. For biotechnologies utilizing nanoconfinement to study DNA, it is necessary to understand internal fluctuations, to ensure that independent samples are being imaged, and to minimize noise or maximize accessible imaging time. Initial measurements by Reisner et al. of DNA relaxation times in nanochannels showed a relaxation time scale on the order of a second, with a local maximum at a transition between two regimes near the Kuhn length.5 In slits, measurements of diffusion and relaxation time by Hsieh et al. show dynamics in between those described by Rouse and Zimm physics.6 In a more detailed study, structural time correlations in slit-confined DNA were examined by Jones et © 2015 American Chemical Society

al., who found that the hydrodynamic exponent of internal correlations grew with spatial separation toward a plateau governed by Zimm physics, before decaying as the separation exceeded the size of the channel, providing experimental evidence that hydrodynamic interactions are screened beyond length scales equivalent to the height of the slit. The internal fluctuation modes of nanoconfined DNA were examined by Karpusenko et al.7 in the context of density variations along stretched molecules, making analogy to standing wave modes on a string. In bulk, a polymer would undergo fluctuations through a large number of Rouse-type modes, each with its own effective spring constant and relaxation time. In a confining system, the free energy potential effectively imposes a global spring constant that dictates the molecule’s fluctuations. This was realized by Karpusenko et al.,7 who showed that nanochannel confinement lead to several apparent standing wave modes. Confining the molecule in a potential landscape is a way of controlling the dominating mode. In complex nanotopographies the longest relaxation time is governed by the time scale for exchange of contour between spatially separated trapping sites. In order to study this phenomenon, we introduce a system of nanofluidic confinement.2 Molecules are partitioned into two adjacent spatially separated cavities connected by a nanofluidic slit (Figure 1a,b). Molecular contour fluctuates from one cavity to the other. Because the molecules are stationary, the internal fluctuations are decoupled from diffusion. Because the reservoirs are Received: May 5, 2015 Revised: June 10, 2015 Published: June 19, 2015 4742

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Figure 1. (a) Schematic showing a side view of the nanocavity system: a cartoon DNA molecule occupies two cavities connected by a slit. (b) Hierarchy of the experimental system: photograph of the chip, optical micrograph of the nanoslits between the two microfluidic reservoirs, and SEM image of the pits. (c) Time-series montage of fluorescence micrographs for a single DNA molecule assembled in the nanocavities, showing fluctuations in intensity between the top and bottom cavity. (d) Time series of the cavity-integrated intensities for three different molecules, showing the anticorrelated fluctuations.

spatially separated, the fluorescent intensity from each can be considered without the effects of diffraction. The double nanocavity system imposes an effective free energy landscape on the confined molecule, leading to a reduction of complexity such that molecular fluctuations can be characterized effectively with only two modes. In particular, the free energy landscape imposes an effective spring constant binding the molecule to each reservoir and an effective spring constant controlling the tension in the molecule. This effectively maps the system to that of a simple coupled harmonic oscillator with only two eigenmodes corresponding to symmetric and antisymmetric oscillations. The time scales associated with molecular partitioning in cavities have largely not been explored. Our earlier work on nanocavity physics focused on static equilibrium physics8 and diffusion.2 Nykypanchuk et al.9 examined DNA fluctuating between two adjacent spherical cavities of different size, focusing on cases where the molecule occupied a single cavity and Cifra 10 modeled the statics of a similar system experimentally in the context of nanopore translocation. Yeh et al.11 examined a DNA molecule confined over a slit between two reservoirs to study the forces acting on the molecule as equilibrium is broken. Kounovsky-Shafer et al.12 also examined long DNA molecules stretched over a nanoslit between two microchannel reservoirs (2 μm deep). They measured the stretching of the nanoslit confined portion of chain and the global transfer time of the molecule across the slit upon retraction of one end of the polymer from a microreservoir. In our system, in contrast to Kounovsky-Shafer et al. and Yeh et al., the nanoscale size of the reservoir cavities enhances the effect of excluded volume so that excluded-volume effects play a key role in determining the relaxation time scales. In addition, we are concerned with fluctuations of contour between adjacent cavities in stable configurations, rather than global transfer of contour across the nanoslit. Beyond merely observing the equilibrium fluctuations of a molecule, we desire to control the fluctuation time scales via modulation of the underlying nanocavity geometry and corresponding free energy landscape.

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twice the depth of the slit (i.e., d = 2h). The devices were fabricated using clean-room procedures described elsewhere.1 The nanofluidic slits were connected with microfluidic reservoirs to macroscopic inlets. λ-DNA stained with YOYO-1 dye at a 10 base pair/dye ratio was diluted from stock in 50 mM Tris with 2% 2-mercaptoethanol (BME) to a final concentration of 1 μg/mL. The DNA solution was pipetted into the chip inlets and the chip was then mounted on a chuck via Oring seals, and the assembly was placed on a fluorescence microscope (Nikon Ti-U). DNA was driven into the nanofluidic slits by applying pneumatic pressure to the reservoirs. The DNA was allowed to equilibrate in the cavities, and then it was filmed with an EMCCD (Andor iXon). The cavities, pits etched in the floor of the slit, offer greater conformational degeneracy and act as entropic traps for molecules, drawing contour into them, while excluded-volume interactions within the cavity push contour back into the slit. Depending on the detailed geometry of the system (e.g., the cavity-to-cavity spacing, cavity width, and depth), a DNA chain may stably occupy one, two, three, or more cavities, residing in a given configuration for minutes at a time without moving (in the bulk, a λ-DNA molecule will diffuse over its own size in less than a second).2,8 Thermal fluctuations occur around these equilibrium conformations, resulting in contour being transferred from one cavity to the next or fluctuating from the cavities into the slit. The fluctuations are correlated: if contour leaves one cavity, it will eventually enter the next. Here we focus on configurations where the DNA occupied two nanocavities (Figure 1a), forming a thermally and mechanically stable structure that persisted for minutes (Figure 1c). Recordings of these molecules were taken, typically of several hundred to several thousand frames, with exposure times typically of 100 ms but ranging from 7 to 200 ms. The intensity in a 5 × 5 pixel square surrounding the intensity peak of each cavity was recorded to form an intensity time series for each cavity (I1(t) and I2(t)). The pixels were roughly 160 nm wide, giving a 800 nm square stencil that was large enough to capture the bulk of the fluorescent intensity but small enough to eliminate artifacts of diffraction from one pit to the other. Typical time series of the fluctuating cavity intensities, shown in Figure 1d, suggest that the cavity intensities are strongly anticorrelated. The large anticorrelation between I1 and I2 implies that the difference in cavity intensities Idiff ≡ I1 − I2 fluctuates with large amplitude (Figure 2b), while correlations in the sum Isum ≡ I1 + I2 are not expected. However, at sufficiently short time scales, there is evidence of time correlation in Isum (Figure 2d), suggesting a faster dynamic mode. Additional evidence for the existence of two relaxation time scales arises from examination of the cross-correlation of the two fluctuating cavity intensities (⟨δI1(0)·δI2(t)⟩, where δI1(0) and δI2(0) represent the deviation from the mean cavity intensity) shown in Figure 2c for an example molecule (note that the raw cross-correlation function is negative; it has been multiplied by −1). Remarkably, we see that the cross-

EXPERIMENTS

The experiments were conducted with a nanofluidic lab-on-a-chip device consisting of a nanoslit with a vertical dimension h on the order of 100 nm embedded with a lattice of square cavities of depth d and cavity-to-cavity spacing S (see Figure 1a). The cavities are etched to 4743

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standard bead−spring polymer model between two entropic traps subject to thermal noise. A standard coarse-grained approach is used to model the polymer.13 Excluded-volume interactions between monomers were implemented by a truncated and shifted Lennard-Jones potential often called the Weeks−Chandler−Anderson or WCA potential.14 This yields a nominal monomer diameter of σ, setting the length scale of the system. The polymer is constructed by joining 100−300 monomers in a linear chain via a finitely extensible nonlinear elastic spring potential in which the constants are set as in the model of Kremer and Grest.15 Semiflexibility was implemented via a three-body harmonic bond with a spring constant of 5kT/σ, which yields a persistence length of ≈5σ, giving similar monomer anisotropy to DNA. The two confining walls of the slit were defined as continuous surfaces with a separation of 2σ between them. The monomer−wall interactions are again given by the WCA potential. The cavities are defined in the lower wall with a depth of 5σ and square in-plane dimensions. Simulations were performed for cavity sizes ranging from 5σ to 16σ and cavity separations of 6σ−16σ. Geometric parameters were not necessarily chosen to match experimental parameters but to ensure stable two-pit occupation. To match the experimental method, the number of monomers in each cavity was calculated by looking down on the cavities and counting the number of beads that have their center within the cavity. Correlations were calculated over 10 000 simulation time units (Figure 3b). The relaxation times were calculated by fitting the exponential decay over the first 10 time-lag steps.

Figure 2. (a) Time series of the integrated intensities in two occupied cavities. (b) Time series plot of the summed two-cavity intensity Isum (black) and two-cavity intensity difference Idiff (red), corresponding to the two intensities in (a). (c) The cross-correlation of the two data sets in (a), the fluctuating integrated intensity between the two cavities. (d) The autocorrelation of the two data sets in (b), the two-cavity intensity sum autocorrelation ⟨δIsum(0)·δIsum(t)⟩ (black) and two-cavity intensity difference autocorrelation ⟨δIdiff(0)·δIdiff(t)⟩ (red) with overlaid exponential fits.

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FREE ENERGY MODEL In order to gain physical intuition into the origin of these scales, we develop a simple free energy model for the cavity system. The relaxation of a confined polymer is given by the ratio of the hydrodynamic drag on the chain, ξ, to the effective spring constant of its local free energy potential,16 κ:

correlation does not follow a single-exponential decay, but instead exhibits two distinct time scales: a short time scale ∼0.1 s and a longer time scale ∼0.5 s.

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SIMULATIONS To confirm that the two time scales represent a universal feature of the confined polymer dynamics, and not specific to our specific experimental model, we perform Langevin dynamics simulations of a coarse-grained polymer in a twocavity system (Figure 3a). As shown in Figure 3c, both the symmetric and antisymmetric modes were reproduced by the simulations. This verifies that it is possible to observe both modes of oscillations in a simple system consisting of a

τ=

ξ κ

(1)

The effective spring constant is the curvature of the free energy potential landscape at equilibrium. To determine κ, we need to find the free energy of the nanocavity system as a function of the contour contained in each cavity. To do so, we modify a previously developed model for predicting the free energy of nanocavity configurations.2,8 Consider a DNA chain of total contour length L straddling two

Figure 3. (a) Sample image from a simulation. (b) Time series of the number of beads in each cavity (blue, green) as well as their sum (black) and difference (red). (c) The autocorrelation function of the summed bead occupancies for each cavity (black) and the autocorrelation of the difference in bead number for each cavity (red). 4744

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Figure 4. (a) Free energy landscape of a chain straddling two cavities as a function of the fraction of contour L1/L and L2/L stored in each cavity. (b) The energy as a function of Ldiff = L1 − L2 and (c) the energy as a function of Lsum = L1 + L2. The forbidden region corresponds to configurations where L1 + L2 + Ls > L. The two potentials in (b) and (c) correspond to the dashed diagonals in the contour plot. The plot was generated by evaluation of eq 6 using physically representative values of A (4.2kT/μm) and B (0.11kT/μm2).

cavities separated by a distance S . Contour L1 is partitioned into the first cavity, and contour L2 is partitioned into the second cavity. The two cavities are connected by a linker strand with length Ls such that L = L1 + L2 + Ls. The free energy of confinement of the chain in this configuration consists of the entropic benefit of contour entering the less confining cavities, the excluded volume cost imposed by stuffing contour in the small cavities, and the free energy of the linker spring Fspring. The entropic benefit of contour entering is proportional to the contour length (L1 or L2). The proportionality factor A represents the entropy gain per unit length of contour entering the cavity. The excluded-volume cost scales quadratically in contour entering with a proportionality factor B = w/Vc, where Vc = da2 is the cavity volume. The total free energy of confinement is then ΔF = −AL1 − AL 2 + BL12 + BL 2 2 + Fspring(Ls) kT

as the total contour in the cavities and the contour difference between the cavities. In terms of these new variables, eq 2 has the form ΔF 1 1 = −ALsum + BLsum 2 + BLdiff 2 + Fspring(L − Lsum) kT 2 2 (6)

Evidently, the free energy decouples into terms purely dependent on Ldiff or Lsum. To demonstrate this point graphically, note the variables L1 and L2 define a free energy landscape with orthonormal coordinates (see Figure 4 which shows a plot of eq 6 for representative values of A and B). The variables Lsum and Ldiff represent diagonal vectors in this coordinate system. When the contour per cavity is equal (i.e., L1 = L2), eq 6 reduces to the equilibrium case considered previously,8 corresponding to a global minimum in the free energy. We can Taylor expand eq 6 for small fluctuations in Lsum and Ldiff away from equilibrium. Define the change in Lsum relative to the equilibrium point via δLsum = Lsum − Leq sum and the change in δLdiff = Ldiff. Then expanding eq 6 to second order, we find

(2)

The simplest parametrization for the spring term is the free energy of an ideal chain: Fspring(Ls) =

3S2 4pLs

with p the persistence qualitatively correct predictions we use a DNA semiflexibilty account:17 Fspring kT

=

ΔF 1 1 = κa(δLdiff )2 + κs(δLsum)2 + O(δLsum 3) kT 2 2

(3)

length. This specification on Fspring gives results, but for more quantitative spring free energy that explicitly takes and height-dependent effects into

S2 Deff − 1 [2LsDeff − S(Deff + 1)] 4pLs 4(Ls − S)

Thus, this system can be treated as a coupled oscillator with two independent modes: a symmetric mode arising from fluctuations of contour in and out of the slit (δLsum varies) and an antisymmetric mode arising from fluctuations of contour from one cavity to the other (δLdiff varies). The antisymmetric mode is purely harmonic, with no higher-order contributions to the potential, while the symmetric terms can be treated as approximately harmonic for small perturbations away from equilibrium. The quantities κs and κa are the effective spring constants for the symmetric and antisymmetric modes, respectively. Comparing eq 6 and eq 7, we observe that κa = B, the excluded-volume parameter. The symmetric mode has an asymmetric potential about equilibrium depending in detail on the precise specification of Fspring. Unlike κa, κs does not have a cromulent closed-form expression in terms of the system parameters.

(4)

Confinement effects are imposed on the elasticity via an 1.441 effective dimensionality, Deff = 1 + 2/(2 − e−0.882(p/h) ), found by measuring in-plane tangent correlations of simulated confined semiflexible chains. The dimer free energy (eq 2) can be conveniently rewritten via a change of variables. Introduce Lsum ≡ L1 + L 2 ;

Ldiff ≡ L1 − L 2

(7)

(5) 4745

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Figure 5. (a) Measurements of antisymmetric correlation time as a function of cavity width compared to a quadratic dependence. (b) Measurements of the antisymmetric correlation time as a function of cavity-to-cavity spacing, for 500 nm (red) and 600 nm (black) wide cavities. Theoretical prediction of free energy model is overlaid. (c) Ratio of the correlation times of the symmetric mode to the antisymmetric mode τs/τa, compared to the free energy model prediction (dashed curve) and the Langevin dynamics simulations (connected circles).

function ⟨δI1(0)·δI2(t)⟩, according to our model, is the sum of the correlation functions for the independent sum and difference modes, explaining why we observe two distinct time scales in Figure 2c. Moreover, if our model is correct, we would expect to see the dependence of τa and τs on device parameters predicted by eqs 9 and 10. Measurements of τa as a function of cavity width (Figure 5a) suggests that τa has a dependence on cavity width consistent with the quadratic scaling predicted by eq 9. Using known geometric values and an effective width w = 9 nm,20 eq 9 predicts an α = 0.48 ± 0.02. Measurements of τa as a function of cavity-to-cavity spacing (Figure 5b) are consistent with the linear dependence predicted by eq 8. The y-intercept of these linear fits can be used to determine ϵ, which we find has a small but non-negligible value (ϵ = 0.3 ± 0.1 for a = 500 nm and 0.2 ± 0.1 for a = 600 nm data). The friction prefactor of roughly 0.5 that we observe is significantly smaller than the 6π deGennes predicted based on the Stokes−Einstein coefficient of a spherical blob. More detailed simulations of similar systems16 and measurements of relaxation in nanochannels5 suggest a friction prefactor on order unity, in agreement with our results. As seen in Figure 2d, the symmetric mode typically relaxes faster than the antisymmetric mode. The ratio of the two relaxation times is, however, highly sensitive to the cavity width (Figure 5c). The spring constant associated with the symmetric mode arises from the curvature of the free energy potential landscape at equilibrium (see Figure 4). This curvature is strongly related to the tension in the linker, calculated using the height-dependent modified Marko−Siggia relationship (eq 4). High tension leads to a high κs. However, as the tension at equilibrium decreases, the curvature of the potential softens, and the spring constant decreases. Moreover, increasing the cavity width pulls contour out of the slit and increases the tension. A small cavity width leads to low tension. Consequently, the behavior of τs as a function of cavity width is opposite that of τa. As the width of the cavities becomes smaller (lower tension), we expect the ratio τs/τa to increase. Experimental measurements of τs/τa are shown in Figure 5c and compared to the predictions of our free energy model. We find decent qualitative agreement, although the theory overestimates τs/τa for small cavity width. Our theoretical model assumes that only the curvature of the two modes differs (i.e., it uses the same friction for both), meaning that the ratio of time scales is given by the function G in eq 10. It is possible that tension reduces the blob length scale to below that of the confinement scale h, giving rise to a tension-dependent friction

In order to deduce the chain friction factor, we introduce a classic blob argument from Brochard-Wyart and de Gennes.18 The friction of a tube-confined polymer scales as the number of statistical blobs in the chain multiplied by the friction per blob, taken as the Stokes drag on a sphere with a diameter equivalent to the blob size. This argument leads to a total linker friction factor proportional to the linker’s stretched length and hence scaling as the cavity-to-cavity spacing S . Thus, the expression for the friction factor is

ξ = αη S + ϵ

(8)

The quantity η is the buffer viscosity. The prefactor α is a numerical proportionality factor, containing the Stokes drag coefficient (classically estimated as 6π) and ultimately to be determined from comparison to experiment. The quantity ϵ represents an offset due to the contribution of the cavities to the total friction. A blob model applied to relaxation in slits is more complicated but yields similar results with the same assumptions. The role of hydrodynamic interactions between segments of the chain in the slit is not fully known, and the measured relaxation times scale between those predicted from Rouse and Zimm scaling. However, the work of Jones et al.19 shows that for the heights considered in our system, below 200 nm, dynamics are effectively Rouse-like. The relaxation time of the antisymmetric mode should then scale as τa =

η d 2 ξ =α a S κas kT w

(9)

No closed-form expression exists for the symmetric mode, but a simplified series expansion can be expressed as τs =

η d 2 ξ =α a S × G (w , p , L , a , S , h , d ) κs kT w

(10)

The function G(w,p,L,a,S ,h,d) is nontrivial and can be found by substituting the value of Lsum that minimizes eq 6 into the second derivative of eq 6 to find the curvature of the symmetric potential about equilibrium.

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RESULTS AND DISCUSSION Our free energy analysis suggests that the fast and slow time scales observed in experiments and simulations arise from the symmetric and antisymmetric transfer modes. We see only one relaxation time scale in the autocorrelation functions ⟨δIsum(0)·δIsum(t)⟩ and ⟨δIdiff(0)·Idiff(t)⟩ because Isum and Idiff represent independent modes. The cavity pair-correlation 4746

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gratefully acknowledges funding from NSERC via a Discovery Grant (2014-06091).

factor. However, the tension blob length scale is typically larger than h; thus, we expect these differences to be insignificant. Because of a wide gap between the experimental and simulation contour length and time scales, it is difficult to precisely map the simulation results to experiment. Instead, the dimensionless mode ratio as a function of cavity size is compared between the simulations and experiments as shown in Figure 5c. We have rescaled the bead diameter by a factor of 52 nm/σ to match experiments. This rescaling includes a factor of 9 nm/σ to match the bead diameter to the effective width and a factor of 5.8 from a simple scaling argument that would ensure two-pit occupancy if the chain were lengthened to match λ-DNA. We find comparable agreement to the data that is seen using the free energy model. In both cases, there is deviation from theory for small cavity sizes, where the symmetric mode is faster than expected. Nevertheless, the trends found in the experimental data, the simulation results, and the thermodynamic model are consistent. From an engineering point of view, these considerations suggest a series of design guidelines for controlling the fluctuation modes. The time scale of the dominant antisymmetric mode can be increased by increasing the size and spacing of the traps, and conversely this time scale can be minimized by shrinking the size and spacing. For fast relaxation and no symmetric mode, large cavities can be placed close together. For fast relaxation with a strong symmetric mode, small cavities can be placed close together. For slow relaxation with a symmetric mode, large cavities can be placed far apart, and for slow relaxation with a strong symmetric mode, small cavities can be placed far apart.

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(1) Reisner, W.; Pedersen, J. N.; Austin, R. H. Rep. Prog. Phys. 2012, 75, 106601. (2) Klotz, A. R.; Brandao, H. B.; Reisner, W. W. Macromolecules 2012, 45, 2122−2127. (3) Han, J.; Craighead, H. G. Science 2000, 288, 1026−1029. (4) Levene, M. J.; Korlach, J.; Turner, S. W.; Foquet, M.; G, C. H.; Webb, W. W. Science 2003, 299, 682−685. (5) Reisner, W.; Morton, K. J.; Riehn, R.; Wang, Y. M.; Yu, Z.; Rosen, M.; Sturm, J. C.; Chou, S. Y.; Frey, E.; Austin, R. H. Phys. Rev. Lett. 2005, 94, 196101. (6) Hsieh, C.-C.; Balducci, A.; Doyle, P. S. Macromolecules 2007, 40, 5196−5205. (7) Karpusenko, A.; Carpenter, J. H.; Zhou, C.; Lim, S. F.; Pan, J.; Riehn, R. J. Appl. Phys. 2012, 111, 024701. (8) Reisner, W.; Larsen, N. B.; Flyvbjerg, H.; Tegenfeldt, J. O.; Kristensen, A. Proc. Natl. Acad. Sci. U. S. A. 2009, 106, 79−84. (9) Nykypanchuk, D.; Strey, H. H.; Hoagland, D. A. Macromolecules 2005, 38, 145−150. (10) Cifra, P. Macromolecules 2005, 38, 3984−3989. (11) Yeh, J.-W.; Taloni, A.; Chen, Y.-L.; Chou, C.-F. Nano Lett. 2012, 12, 1597−1602. (12) Kounovsky-Shafer, K. L.; Hernández-Ortiz, J. P.; Jo, K.; Odijk, T.; de Pablo, J. J.; Schwartz, D. C. Macromolecules 2013, 46, 8356− 8368. (13) Slater, G. W.; Holm, C. E. A. Electrophoresis 2009, 30, 792−818. (14) Weeks, J. D.; Chandler, D.; Andersen, H. C. J. Chem. Phys. 1971, 54, 5237−5247. (15) Grest, G. S.; Kremer, K. Phys. Rev. A 1986, 33, 3628. (16) Tree, D. R.; Wang, Y.; Dorfman, K. D. Biomicrofluidics 2013, 7, 054118. (17) de Haan, H. W.; Shendruk, T. N. ACS Macro Lett. 2015, 4, 632−635. (18) Brochard, F.; de Gennes, P.-G. J. Chem. Phys. 1977, 67, 52−56. (19) Jones, J. J.; van der Maarel, J. R.; Doyle, P. S. Phys. Rev. Lett. 2013, 110, 068101. (20) Stigter, D. Biopolymers 1977, 16, 1435−1448.

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CONCLUSIONS We have performed correlation-based measurements of contour exchange within a single DNA molecule straddling two nanofluidic cavities. The imposed free-energy landscape effectively reduces the dynamics to that of a coupled harmonic oscillator with contour exchange arising between a symmetric and antisymmetric mode. Our antisymmetric relaxation time measurements agree well with a simple free energy model and a friction factor that scales linearly with the cavity spacing. Measurements of the ratio of time scales between the two modes gives results corroborated by our model as well as Langevin dynamics simulations. Future work will examine higher-order states in greater detail and fluctuations in chains of varying topology (e.g., branched and circular).

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REFERENCES

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (A.R.K.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by a National Science and Engineering Research Council of Canada Discovery Grant (NSERC-DG, 386212-10) with equipment provided by an NSERC tools grant and the Canada Foundation for Innovation (CFI). In addition, the authors thank Laboratoire de Micro- et Nanofabrication (LMN) at INRS-Varennes and McGill Nanotools-Microfab for supporting the nanofabrication. We thank the facility for electron microscope research at McGill (FEMR) for providing access to the scanning electron microscope. HdH 4747

DOI: 10.1021/acs.macromol.5b00961 Macromolecules 2015, 48, 4742−4747