Article pubs.acs.org/Macromolecules
Diffusion Resonance of Nanoconfined Polymers Alexander R. Klotz,* Hugo B. Brandaõ , and Walter W. Reisner Department of Physics, McGill University, Montreal, Quebec, Canada H3A 2T8 ABSTRACT: We examine the diffusive behavior of single polymers under spatially varying entropic confinement. A nanofluidic slit embedded with a lattice of pits was used to constrain single DNA molecules to discrete conformational states. Diffusion was characterized by dwelling in specific conformations followed by transitions to neighboring states. In contrast to studies involving simple 2D (nanoslit) and 1D (nanochannel) geometries, the diffusivity showed nonmonotonic dependence with respect to the parameters of confinement. In particular, the nanopit array allows us to fine-tune the diffusivity of a single polymer to a local resonance minimum. Moreover, we show that energetically favorable states dominate over higher energy states and that a single state can be stable over a wide range of parameter space. These stable states correspond to resonances in the diffusion.
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INTRODUCTION Over the past decade workers at the interface of polymer physics, nanoscience, and biotechnology have developed the field of DNA nanoconfinement, focused on using nanoconfined geometries to modulate DNA conformation on-chip. The bulk of work in the field has been focused on simple confinement, e.g., two-dimensional nanoslits and one-dimensional nanochannels.1 In simple geometries, the confinement alters the DNA conformation, leading to either a pancake-like conformation (nanoslit) or an extended cigar-like conformation (nanochannel). Highly parallel nanochannel arrays can be used to extend many DNA molecules in parallel and forms the basis of new type of single-molecule mapping technology.2,3 There has been significant work investigating the diffusion and equilibrium conformation of DNA confined in nanoslits and nanochannels.4−9 In these simple devices polymer dynamic and static properties are observed to scale monotonically with confinement (although controversy still exists regarding the details of the scaling exponents10,11). Nanoslit and nanochannel geometries allow only for a uniform expansion of the polymer coil, either along the axis of the nanochannel or perpendicular to the confining lid of a nanoslit device. Recently, Reisner et al.12 proposed a new type of complex nanodevice consisting of an open nanoslit with embedded nanotopography. By spatially varying confinement, it is possible to create a user-defined free energy landscape for controlling polymer conformation. Polymers will have greater conformational freedom in locally less confined regions of the device, leading to a locally increased entropy. A higher local entropy corresponds to a free energy minimum, acting as a potential trap for molecule contour. In particular, if the confinement is varied over scales comparable to the molecular gyration radius, it is possible to directly “sculpt” the conformation of a single DNA molecule. In the initial study of Reisner et al.,12 the embedded nanotopography consisted of © 2012 American Chemical Society
a square lattice of nanopits. The pit depth, pit-to-pit separation, and slit height could be adjusted via the fabrication process. This device demonstrated enhanced control over local polymer conformation, in particular the ability to direct self-organization of DNA into complex conformations linking multiple nanopits, demonstrating simultaneous stretching and trapping of DNA. This system provides an ideal platform for dictating polymer self-organization in a solid-state device, and the templated selfassembly that it facilitates may reduce the need for highly specific surface chemistry. We show here that the nanopit arrays can be used to control dynamic properties such as single-molecule diffusion, in addition to static equilibrium conformation. We explore a wide enough range of parameter space to deduce the detailed dependence of diffusion and static conformation on device geometry. We show that, in contrast to simple geometries, where the diffusion depends upon confinement in a purely monotonic way, the diffusion has a complex nonmonotonic dependence, undergoing a resonance effect as a function of the nanopit width. The original work of Reisner et al.12 focused purely on static properties of DNA molecules in the nanopit lattices. Reisner et al.,12 in particular, did not demonstrate the existence of stable conformational states over a range of device parameters (as predicted by their model): the four lattice spacings investigated were not sufficient to demonstrate the existence of stable conformations (corresponding to plateaus in the pit occupancy versus lattice spacing). Subsequent work examined the mobility of DNA in linear nanopit arrays, both experimentally13 and theoretically.14 Again, these authors did not explore a wideenough range of parameter space to make quantitative Received: December 2, 2011 Revised: February 7, 2012 Published: February 15, 2012 2122
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Figure 1. (a) Schematic of the overall design of the device. (b) Schematic of the nanoslit and embedded nanopit lattice with DNA and fluorescence images of molecules in the device. Scale bar is 2 μm. Typical dimensions: slit height (h): 100 nm; pit width (a): 500 nm; pit separation (l): 1000 nm.
Figure 2. Free energy landscapes as a function of contour partitioned per pit. Each curve, corresponding to a given occupancy state on the lattice (with N occupied pits), gives the free energy as a function of the contour partitioned per pit (expressed as a percentage of the total molecule contour). The curve minima correspond to the equilibrium values of contour partitioned per pit. When the free energy differences between the curve minima are much greater than kbT, the molecules will be found in a single state. When the free energy differences are on the order of kbT, multiple occupancy states will be accessed in equilibrium. In (a), the N=3 state dominates, but transitions are also possible to N=2 and N=4 states (although of low probability). In (b), the N=3 state still dominates but the N=4 and N=2 states are more probable. The devices were fabricated using standard clean-room techniques. The nanopits were patterned onto a fused silica wafer in zep520A resist using electron beam lithography and etched using CF4:CHF3 reactive ion etching. The nanoslit and microchannels were patterned in S1813 resist using UV lithography and etched similarly. The individual wafers were bonded to coverslips to seal the slits and channels. λ-DNA was used as a model polymer in these experiments. After being stained with YOYO-1 fluorescent dye with a 10:1 intercalation ratio, it had a total contour length of ∼19 μm and a persistence length of ∼52 nm.18 The stained DNA was loaded into a 50 mM TRIS buffer (pH 8.0) with 2% β-mercaptoethanol to prevent photobleaching. The DNA was pipetted into the reservoirs of the chip, and the chip was fastened to a chuck and attached to a Nikon Eclipse TI inverted microscope. A Nikon Plan Apo VC 100× lens was used to focus on the system. By applying pneumatic pressure externally to the reservoirs, DNA could be driven from the microfluidic channels into the nanoslit. When the driving pressure was turned off, the molecules quickly selfassembled into the configurations dictated by the nanopits and a time series was recorded using a cooled Andor iXon EM-CCD camera. The images were analyzed by tracking the position of each occupied pit over time.
predictions for the dependence of mobility on device parameters. Moreover, the two-dimensional lattice behavior discussed here offers a richer experimental landscape than the one-dimensional arrays. Additional work by Mikkelson et al.15 is similar to that of DelBonis-O’Donnell,13 in that the mechanisms of trapping and escape under external driving are examined, but the effects of varying confinement dimensions and equilibrium behavior are left unexplored. Finally, our system differs from classic work on entropic trapping of DNA, such as that of Han and Craighead16 or Nykypanchuk et al.,17 in that it involves strong lateral confinement as well as vertical confinement, leading to the ability to fine-tune local DNA conformation.
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EXPERIMENT
These experiments took place in a nanofluidic lab-on-a-chip device (Figure 1), first described by Reisner et al. in 2008.12 A nanoslit, on the order of 100 nm in height and smaller than the radius of gyration of the polymer, constrained the dynamics of the polymer to two dimensions. The nanoslit was embedded with a square lattice of square pits with double the depth of the slit, on the order of a few hundred nanometers to a micrometer in width with separations on the order of a micrometer. Because of the greater conformational degeneracy within the pits, the square pits acted as entropic traps.
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THEORY In this network of pits, entropy is maximized when the entire polymer occupies a single pit, but self-exclusion interactions 2123
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Figure 3. Static behavior of DNA molecules in this system: (a) A plot of average occupancy as a function of pit size in a 50 nm slit, with theoretical fit. Occupancy decreases with increasing pit size. Fluorescence micrographs show the corresponding configurational states. Scale bar is 2 μm. (b) A plot of average occupancy as a function of pit size in a 70 nm slit, with theoretical fit. A plateau is apparent for the dimer state. Error bars represent variation between molecules, and may be absent if all molecules were observed in a single state.
Figure 4. Dynamic behavior of molecules in this system. (a) The mechanism of diffusion, involving discrete hops between energy states that displace the center of mass. (b) Two model free energy landscapes demonstrating that diffusion can be fast or slow depending on the stability of each state. (c) Time series of a DNA molecule demonstrating slow and fast diffusion.
scaling laws or fit to experimental results. This theory is used to calculate the probability of each occupancy state N for a given set of device parameters. The theory predicts a decreasing mean occupancy for increasing pit size. It also predicts plateaus, regions of parameter space over which a single state dominates (Figure 3). The dynamics of a polymer in a fluid under equilibrium are described by diffusion. Single polymer diffusion is characterized by Rouse−Zimm dynamics in free solution and by reptation in a highly disordered medium.21 In a nanoslit, polymer diffusion is described by a chain of “blobs”, where chain segments within each blob behave as if free. Hydrodynamic interactions between blobs are screened,5 leading to a − 2/3 power scaling law of diffusion with respect to slit height. Numerical simulations from Hickey and Slater22 show a nonmonotonic dependence on diffusion between the Rouse and reptation disorder extremes. Experiments by Nykypanchuk et al.17 suggest that entropic trapping leads to a distinct polymer diffusion mechanism. This entropic trapping regime is characterized by a polymer dwelling in the trap and escaping in discrete steps, analogous to the diffusion mechanism investigated here. The mechanism of diffusion in this system is as follows: a DNA molecule occupies N pits. The molecule becomes thermally activated or relaxes and transitions to another state and now occupies N − 1 or N + 1 pits (Figure 4), leading to a displacement of the molecule center of mass. Typically, a
arising from binary chain collisions in the restricted pit volumelimit the contour that can be stored in a single pit. Thus, a single polymer may occupy multiple pits. These competing interactions dictate a free energy landscape, with the amount of contour in each pit acting as the independent variable (Figure 2). At equilibrium, the Helmholtz free energy of the polymer is determined entirely by the number of occupied pits. The absolute free energy of each state can be tuned by varying geometric parameters such as the height of the nanoslit or spacing of the pits, but ultimately the molecule is in a system of energetically discrete configurations where the number of occupied pits is the only dynamical variable. The simplification of the system to one variable simplifies it greatly, both experimentally and theoretically. The free energy for a molecule occupying N pits is ΔF(N ) = N (−AL p + BL p2) + (N − 1)Fs k bT
This model is derived by Reisner et al.12 and elaborated upon by Klotz.19 Lp represents the amount of contour in the pit, determined by minimizing the free energy. Fs represents the free energy of the spring linker spanning adjacent pits.20 A and B are prefactors that represent the strength of the entropic and excluded volume interactions, respectively. B is inversely proportional to the volume of each pit. They may be defined based on geometric definitions assuming certain free energy 2124
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Figure 5. Measurements of diffusion with respect to pit width for two systems, alongside occupancy measurements. (a) Diffusion and occupancy as a function of pit width in a 100 nm nanoslit. A local minimum in diffusion corresponds to a stable dimer state. (b) Diffusion and occupancy as a function of pit width in a 70 nm nanoslit. Two local minima in diffusion corresponds to stable dimer and trimer states.
system that exists on a plateau, the molecules are expected to be found predominately in a single state, suggested by a nearinteger mean occupancy. In addition, relatively few transitions to higher or lower states are expected, corresponding to the aforementioned diffusion resonances. Plateaus were observed in several devices corresponding to a stable N = 2 state. The experimental observation of the plateaus suggests that the nanopit−nanoslit system can be used to create stable conformations at equilibrium.
molecule will tend to be found in its lowest energy ground state. It becomes thermally activated to the first excited state, where it decays to its ground state. Over longer periods of time, a molecule will diffuse throughout its environment by hopping between different occupancy states. The lifetime of a ground of order several minutes, while a higher energy state may last tens of seconds; the lifetimes of states are orders of magnitude greater than the Rouse relaxation time of polymers in the pit.21 The characteristics of this mechanism can be examined with a toy model: a polymer on a one-dimensional array with two accessible states (a single pit state and a two pit state; Figure 4). Assuming a certain rate R12 for the transition from the one to two pit state and a rate R21 for the backward transition, it can be shown that the effective rate for motion of the molecule centerof-mass one position over on the lattice is R = R12R21/(R12 + R21). The net rate is maximized when R12 = R21, corresponding to a system in which each state is of equal probability (e.g., via detailed balance). If the molecule spends a long time in either the one pit or two-pit state, corresponding to a system in which one state is of much lower energy than the other, the net rate will be strongly diminished. If only one state is accessible, motion is not possible. However, a more complex model is needed when a diffusing molecule can access multiple occupancy states on a square lattice. In this case there is no analytic expression for the diffusion constant as a function of the transition rates.
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DYNAMICS While it is known that slit confinement affects the diffusivity of a polymer,5 these experiments show that an additional level fine-tuning can be achieved with embedded nanotopography. In weakly bound systems where the free energy difference between states is relatively small, there is a monotonic decrease in diffusivity when the pit size is increased: in a 200 nm nanoslit with 1 μm pit spacing, the diffusion coefficient in a system with 1 μm pits was a factor of 90 smaller than that in a system of 350 nm pits. This allows an additional degree of freedom in the fine-tuning of the diffusivity of a single polymer, beyond confinement due to slit height. In systems with greater energy separation of their states, the diffusion shows nonmonotonic behavior. Diffusion resonances were observed, as a function of nanopit width, in devices with a slit height h = 70 and 100 nm and lattice spacing l = 1000 nm (Figure 5). Both systems showed a diffusivity minimum for an occupancy N = 2 state and a corresponding diffusion maxima at occupancy values between 1 and 2. In particular, for the 70 nm system, multiple occupancy minima were observed (at N = 2 and N = 3 states, with corresponding maxima between occupancy values of 1 and 2 and 2 and 3). Local minima exist when the molecule is strongly bound to a certain state because there are fewer thermal transitions to higher or lower states. These local minima occur at integer values of the mean occupancy (because only one occupancy state is accessible for these parameter ranges). Such a resonance can be seen in Figure 5, where local minima in diffusion correspond to integer values of mean occupancy. The requirement for diffusion minima is that the ground state is much lower in energy compared to the first and second excited states (Figure 4). In systems with higher occupancy ground states, for example those with smaller pits, the excited states tend to be closer together in energy. Thus, it is difficult to observe diffusion minima at, for example, the N = 4 state. This
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STATICS The diffusion of polymers in this system is best discussed in the context of their static behavior. As the size of the pits was varied, the equilibrium state tended to change, as seen in Figure 3. In systems with the largest pits the molecules occupied only a single pit: the monomer state. As the pits decreased in width, the molecule tended to occupy two (dimer) and then three and four pits. Increasing confinement leads to stronger excluded volume effects, forcing the molecule to occupy more pits, while decreasing confinement allows entropic effects to dominate. By measuring the mean occupancy of an ensemble of polymers as a function of pit width for a single slit depth and lattice spacing, the behavior of the system with respect to pit size was observed (Figure 3). The behavior with respect to lattice spacing was discussed by Reisner et al.12 As described qualitatively, the mean occupancy tended to decrease with increasing pit size. Theory predicts regions of parameter space over which a single state dominates, termed plateaus. Within a 2125
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to trap a biomolecule in a partitioned state. A device consisting of an array of nanopits (termed nanowells) in a gold surface has been used as a biosensor,24 an application that is also possible with the nanofluidic devices in these experiments. While we have mainly focused on the effect of varying geometry using λDNA, it is also possible to use a fixed geometry with varying sized molecules. In that scenario, molecules whose lengths are resonant to that particular geometry will become trapped and will diffuse much more slowly, so that the nanopit structures will act as a passive trap for molecules of specific lengths. A similar system was described by Nykypanchuk et al. in a 2002 paper.17 DNA was observed in a two-dimensional network of hollow spheres of submicrometer diameter connected by holes on the order of 100 nm in diameter. Some of the features of DNA in a network of traps are observed in both the nanofluidic and spherical systems: stable occupation of multiple traps and constrained diffusion due to discrete hopping between traps. In a subsequent thesis,25 a nonmonotonic behavior of diffusivity was reported but not understood. In addition to the nonmonotonic diffusivity dependence on geometry, stable higher-occupancy states were observed, with a free energy minimum corresponding to the dimer state. Both these effects, while unexplained by Nykypanchuk and colleagues, serve to demonstrate the main point of this paper: that diffusion is minimized when a polymer is entropically trapped in a stable state. In conclusion, the experiments presented in this paper indicate that a nanopit−nanoslit system can be use to fine-tune the diffusivity of a single polymer to a resonance minimum or maximum. The positions of the diffusion minima correspond to plateaus in the occupancy (where only a single state is accessible). The positions of the diffusion maxima correspond to parameter combinations where a number of occupancy states are accessible, allowing the DNA molecules to diffuse via successive transitions between different occupancy states. We hope to explore several questions in additional work. In particular, one issue highly relevant to understanding how longer molecules (occupying many pits) would behave in the nanopit arrays is whether pit-to-pit transitions that occur at the end of a molecule follow a different mechanism than transitions which occur via herniation from the molecule midsection. Such a question could be explored by experiments with circular DNA (following the example of previous work26). Finally, it would be interesting to probe thermal fluctuations of contour between the pits and linkers via stroboscopic imaging with a pulsed laser. While this study has focused on diffusion, it should be possible to extend the concept of diffusion resonance to transport under constant driving forces (such as hydrodynamic flow13,15 or electrophoresis). The dynamic mechanism discussed here motion mediated via transitions between neighboring occupancy statesshould apply as well if the dynamics are driven by a constant force, and the force is weak enough (or the vertical confinement strong enough) so that molecules remain assembled in the pits during the motion. In this case, molecular transport would be maximized at the same parameter values that give rise to a diffusion resonance. We predict that the nanopit structures would in effect be “transparent” to molecules only in a certain size range (blocking passage of molecules in other ranges), creating a molecular “band-gap” effect.
can be overcome by placing the system in a narrower slit, which increases the energy difference between states. Unfortunately, it is experimentally difficult to observe diffusion in these systems because the occupancy times of the stable states can last over an hour. Molecules of the same size are expected to have the same dynamics, but a dispersity in sizes would lead to molecules with different equilibrium behavior and different diffusion coefficients. λ-DNA is known to cleave near its midpoint due to an AT-rich block;2 analyzing the intensity and average occupancy of each molecule allowed us to remove fragmented DNA from the analysis. If a small pressure imbalance exists in the microchannels, a net hydrodynamic flow will exist and the dynamics would not be purely diffusive. Data in which the direction of diffusion was clearly biased in one direction along the nanochannel were excluded. One final issue can lead to systematic errors in the measurements: it is critical to ensure that the distribution of pit states sampled actually reflects the equilibrium distribution. In particular, metastable states can exist at certain parameter values:12 these are long-lived states that are not accessible from equilibrium states. Metastable states are formed initially when the molecules are introduced into the nanopit arrays. Including metastable states in the analysis will lead to an overestimation of the average occupancy and also the diffusion constant (i.e., a decay from a metastable to equilibrium state could be confused with a single diffusive lattice step). Metastable states can be identified experimentally if transitions between occupancy states proceed almost completely in one direction. In our data, metastability is a problem for only one parameter value (in Figure 5a): the point taken for a 750 nm pit width. In particular, the dimer state for a 750 nm width is metastable: decays to single-pit states are observed with very few backward transitions (10 times as many decays from a two-pit to single pit state versus the one-to-two transition). Average occupancy in this case was estimated by detailed balance. The relative abundances of observed transitions was used to estimate the ratio of the state probabilities P2/P1 = R12/R21. The normalization condition P1 + P2 = 1 could then be used to extract the absolute state occupation probability (once the ratio was determined). The diffusion constant for the 750 nm data point was calculated using only the molecules that were in the N = 1 state at t = 0.
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DISCUSSION The vanishing diffusion in the N = 1 state is important as it indicates that the nanopit devices can be used to localize polymers. From the point of view of our model, this result follows simply because in the limit of large pit separations only one occupancy state is accessible. In this sense, single pits can trap molecules for tens of minutes, and molecules can also be held in higher occupancy states for similar time scales. Electrostatic trapping in nanoscale pits is an alternative to optical tweezers,23 and entropic trapping in a single pit could be used as a method of studying biomolecules, as evidenced by the long occupation times of the monomer state in these experiments. Entropic trapping has benefits over double-layer electrostatic trapping, in that it has much more flexibility in terms of the dielectric properties of the trapped particle, as well as the range of salt concentrations that are necessary to achieve efficient trapping. We also observed dimer states persisting for over half an hour without transition, suggesting such a system can be used
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. 2126
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Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work was funded through the Natural Sciences and Engineering Research Council of Canada (NSERC) discovery grants program (RGPIN 386212). We thank Henrik Flyvbjerg for illuminating discussions and Philippe Fortin Simard for his assistance. We also gratefully acknowledge use of the DANCHIP fabrication facility (at the Technical University of Denmark), the Nanotools fabrication facility (McGill University), and the Laboratoire de Micro et Nanofabrication (at l’institute nationale de recherche scientifique-INRS, Varennes Quebec).
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REFERENCES
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