Nano-plumbing with two dimensional metamaterials

Abstract

Complex manipulations of DNA in a nanofluidic device require channels with branches and junctions. However, the dynamic response of DNA in such nanofluidic networks is relatively unexplored. Here, we investigate the transport of DNA in a two-dimensional metamaterial made by arrays of nanochannel junctions. The mechanism of transport will be explained as Brownian motion through an energy landscape formed by the combination of the confinement free energy of DNA and the effective potential of hydrodynamic flow, which both can be tuned independently within the device. For the quantitative understanding of DNA transport, we propose a dynamic mean-field model of DNA at a nanochannel junction. We show that the dynamics of DNA in a nanofluidic device with branched channels and junctions is well described by our model.

Confined polymers have long been theoretically described,^[1–4] and experimental as well as computational studies have largely confirmed those predictions.^[5–12] DNA, often studied as the prototypical polymer molecule, extends along the axis of confining channel if the size of channel is less than the radius of gyration of the molecule.^[1,13] The extended molecule can be manipulated for various applications, such as DNA sorting,^[14,15] gene mapping,^[16–19] single molecule experiment,^[20,21] and fundamental polymer physics experiment.^[22,23] The physical properties of a confined polymer depend on the contour length (L), the effective width (w), the Kuhn length (l_k ), and the size of the confining channel (D). Based on the ordering of these length parameters, the physical properties of a confined polymer in nanochannel fall into different regimes. The classic de Gennes regime^[2] and the Odijk regime^[3] pertain to two extreme ends of confinement, where $D\gg l_k$ and $D\ll l_k$ respectively. The equilibrium properties of a confined DNA molecule, such as extension, variance of extension, and the confinement free energy are well established in these two extreme regimes.^[24–26] However, the transition between these two extreme regimes are still subject to some discussion.^[27–31]

The dynamics of confined DNA in a nanochannel plays a vital role in many nanofluidic applications^[32,33] since the timescales of thermal fluctuation dissipation are comparable to the measurement time.^[34] So far most of the studies have considered linear and uniform channels. However, complex manipulation of DNA requires gradient and branched channels of different sizes.^[35–37] The dynamics of DNA at the junction of such branched channels are influenced by entropic forces generated due to differences in channel sizes and herniation.^[38–40]

We believe that such junctions could play a crucial role in manipulating DNA that is already stretched, enabling single molecule investigations of interactions of polymers under confinement as different buffers flow over it.^[41–43] We perceive a shortcoming in all of these approaches that use the movement of the buffer over a stationary molecule in as far as it typically involves switching times that are diffusion limited and require fine control. An alternative path has been explored for larger entities in microfluidic devices such as cell, nuclei, etc., which is to displace the biological molecule relative to the flowing stream.^[44–52] These devices are often designed as anisometric arrays of structures that have different transport responses for buffers and molecules. However, these devices are typically size dispersive for DNA translocation, and release DNA from confinement so that a real-time response cannot be determined.

Here we present a composite structure formed by a fluidic network of nanochannels. The network creates a metamaterial with engineered mean transport properties for DNA molecules such as anisotropic mobilities that are dependent on the driving strength, and an effect reminiscent of ballistic transport. We show the large-scale translocation of DNA with three distinct transport modes in the device that are controlled by the applied driving pressure. These transport modes will be explained by the dynamics of DNA at a channel junction which is function of the confinement free energy of DNA and quasi-potential of hydrodynamic flow in the channels, which both could be tuned independently in the device. We present a mean-field dynamical model of DNA to illustrate quantitatively the dynamics of DNA at a junction, and use it to simulate the translocation of DNA through the entire device at various flow speeds.

The design of our nanofluidic device is shown in Figure 1. The metamaterial region is made of arrays of nanochannels that meet at 3-way junctions. The fluidic network consists of three types of nanochannels. ‘Undulating channels’ of size 140×120 nm² (width×depth) connect two microchannels. Adjacent undulating channels are connected by channels of two different sizes: ‘wide link channels’ of size 180×120 nm² (width×depth), and ‘narrow link channels’ of size 115×120 nm² (width×depth). Pressure in the range of 0–1.45 psi is applied between microchannels to drive DNA transport in the metamaterial region. Link channels have no fluid flow due to the isobars generated by the homogeneous flow of fluid in the undulating channels.

Figure 1.

Schematic of the nanofluidic device. A. An overview of the device chip (not drawn to scale) showing the center nanofluidic region (90×90 μm²) which is connected to the reservoirs by the microchannels (100 μm width × 1 μm depth) at two ends. C. The schematic of metamaterial region showing the fluidic networks. The undulating channels of cross section 140×120 nm² (width × depth) run vertically between microchannels, and adjacent undulating channels are connected by the link channels of length 1 μm. The scalebar is 10 μm. B. A zoomed-in schematics of nanofluidic region. The link channels are of two different sizes: wide link (blue) channels of cross section 115×120 nm² (width × depth), and narrow link (red) channels of cross section 180×120 nm² (width × depth). The wide and narrow link channels alternate vertically, and the same size of link channels run diagonally. D-F. SEM image of D. a section of nanofluidic region. E. the narrow link channel. F. the wide link channel.

The device was fabricated in a fused silica substrate. Nanochannels were patterned using electron beam lithography, and the microchannels were patterned using optical lithography. All channels were etched by reactive ion etching. Finally, channels were sealed by fused silica coverslip using thermal bonding.^[53]

We used λ-phage DNA (48.5 kbp, unstained contour length 16.5 μm, New England Bio-labs) stained with bis-intercalating YOYO-1 dye (Life Technologies) at 1:10 molar ratio of dye to base pairs. TBE buffer at 0.5x concentration was used for the suspension of DNA in channels. DNA molecules were observed using inverted fluorescent microscope (Nikon, TE-2000) with 100x (NA=1.35) and 60x (NA=1.40) oil immersion objective lenses under 488 nm laser excitation. Movie frames were illuminated with 10 ms flashes, and recorded using EMCCD camera (iXon, Andor) at 7 to 11 frames per second. (Detailed experimental conditions in Supporting Materials).

For each recording of DNA transport in the metamaterial region, the molecules were pushed from a microchannel into the undulating channels using high pressure (≈15.0 psi between two ends of the microchannels). Once a molecule was in an undulating channel, it was driven at a constant pressure in the 0–1.45 psi range. We mapped the applied pressure to flow speed of buffer by measuring the speed of DNA molecules in a region of the undulating channel without junctions.

Figure 2 illustrates the transport of DNA in the nanofluidic region at various flow speeds (movies in Supporting Materials). We observe three different periodic transport modes of DNA. Each transport mode is identified by a ratio of the number of links to the number of undulating segments (between junctions) that DNA travels within one period of the motion. A period of motion is defined by a repetitive pattern of DNA transport through the link and undulating channel segments. In MODE 1/1 (Figure 2A, 2D), DNA travels through a link channel and a segment of undulating channel during one period of the trajectory. Similarly, in MODE 1/3 (Figure 2B, 2E) DNA travels three undulating channel segments before entering a link channel. In MODE 0/1 (Figure 2C, 2G), DNA does not enter any link channels. These transport modes of DNA can be characterized by either their instantaneous molecular configuration or the mean direction of travel.

Figure 2.

Images illustrating the motion of DNA. A-C. Time-summed images of movie frames for single molecules. The inset images show an instantaneous configuration of a DNA in motion taken during a single 10 ms exposure. A. Motion in MODE 1/1 at flow speed ≈0.5 μm/sec. B. Motion in MODE 1/3 at flow speed ≈0.5 μm/sec. C. Motion in MODE 0/1 at flow speed ≈6.0 μm/sec. D-E,G. Schematics illustrating the motion of DNA in a small region of the device. D. Motion in MODE 1/1, as seen in panel A. E. Motion in MODE 1/3, as seen in panel B. G. Motion in MODE 0/1, as seen in panel C. F. Schematic of the configuration angle (θ) of a DNA molecule.

We identify the instantaneous configuration of DNA by an angle of orientation of the molecule relative to the average flow axis of the undulating channels (Figure 2F). For a single frame with pixel intensities $I\left(x,y\right),$ where x, y are pixel positions, $\left(p,q\right)$-central moments are defined as ${\mu }_{pq}={\sum }_x{\sum }_y{\left(x-\overline{x}\right)}^p{\left(y-\overline{y}\right)}^{q}I\left(x,y\right).$ Here $\overline{x}$ and $\overline{y}$ are the centroid coordinates of the molecule. Using the central moments, we calculate the configuration angle as

$$\theta =\frac{1}{2}arctan\left(\frac{2{\mu }_{11}}{{\mu }_{20}-{\mu }_{02}}\right)$$ (1)

For the direction of DNA transport, we calculate an angle, named the drift angle, between the displacement vector of centroids of DNA for two consecutive frames of a movie and the average flow axis of the undulating channels.

In Figure 3 we illustrate how the different transport modes of DNA manifest themselves in the configuration angle. For a single recording of DNA transport in MODE 1/1, the average value of the configuration angle is ≈55.0° (Figure 3A), while it is ≈6.0° for MODE 0/1 (Figure 3B). Interestingly, the histogram for MODE 1/3 has two peaks with average values of the configuration angle ≈4.0° and ≈51.0° (Figure 3C). This is due to the presence of two distinct configurations of DNA in MODE 1/3 (Figure 2B(i-ii)), and these two configurations undergo an oscillatory motion (Figure 3C(i)). We identify this transport mode by time averaging the configuration angle over the period of oscillation (≈30.0°).

Figure 3.

A-C. Histograms of configuration angle for individual molecules. A. MODE 1/1 at flow speed ≈0.5 μm/sec (see Figure 2A). B. MODE 0/1 at flow speed ≈6.0 μm/sec (see Figure 2C). C. MODE 1/3 at flow speed ≈0.5 μm/sec (see Figure 2B). The figure in the inset shows an oscillation of the configuration angle in MODE 1/3. D. Plot of the configuration angle at various flow speeds for all the recordings of DNA transport (≈300 movies of individual molecule totaling ≈50000 frames). The scalebar is the number of frames. E. Plot of the drift angle vs flow speeds shows three transport modes. F. Plot of the drift angle vs the configuration angle.

We now are able to correlate the configuration angle, flow speed, and drift angle. The histogram of average configuration angle versus average flow speed for all recorded molecules shows three distinct clusters associated with the three modes of DNA transport (Figure 3D). At low flow speed (≲ 1.0 μm/sec) DNA transports either as MODE 1/1 or MODE 1/3. However, as we increase the flow speed to ≳ 4.0 μm/sec, DNA only transports as MODE 0/1. The histogram for the drift angle versus the flow speed similarly shows three clusters associated with each of the transport modes (Figure 3E). We confirm the correlation of the drift angle and the configuration angle in Figure 3F, which shows that the local orientation (configuration angle) of a DNA molecule maps directly to the transport direction (drift angle) of the molecule. This correlation between the local configuration, transport property, and flow speed suggest that all three can be linked by a single model. In the following we develop a model based on the local molecular configuration of DNA at a single nanochannel junction.

The basis of the model is the balance between the confinement free energy of DNA and the quasi-potential generated by hydrodynamic flow in the channel. The confinement regime applicable here $(l_k\ll D\ll l_k^2/w)$ is commonly known as the extended de Gennes regime.^[6,7,54] Functionally, the free energy of confinement scales linearly with the equilibrium extension of a DNA molecule along the channel. In the supporting materials we outline that we treat singly-occupied channel segments according to a recent treatment by Dai et al.,^[55] while doubly-occupied channel segments are treated by correctly scaling the entropic and excluded-volume contributions.

The effect of the buffer medium and its flow is obtained by assuming a viscous damping fluid which results in viscous forces that far exceed inertial forces. Azad showed that DNA transport could be described by a parabolic quasi-potential of fluid flow combined with a viscous force that takes the form of motion relative to the channel walls.^[43] The specific form is shown in the Supporting Materials.

Now consider a DNA molecule arrives at a nanochannel junction while flowing in an undulating channel (Figure 4A). At the junction, DNA will spontaneously form a small loop due to thermal fluctuations, commonly known as a hernia. The hernia formation can be either in the downstream channel or in the link channel (Figure S1). In our experiments, we observed the hernia formation in the downstream channel in ≈73% of all junction-molecule interactions, compared to only ≈10% in the link channels at low flow speed (≲3 μm/sec). The rest of interactions result in more complicated molecular topologies. Therefore, we consider only the model with hernia in the downstream channel (Figure 4B). We could similarly consider a model with hernia formation in the link channel (Figure S1F) or the other molecular topologies, but those results would contribute to the ensemble results only in a small portion.

Figure 4.

A. Schematic of a section of nanofluidic region. B. Model of a DNA at the junction with hernia in downstream channel. Three channels of the junction are labeled as the upstream, downstream and link channels based on the fluid flow direction. The upstream and downstream channels are of the same sizes (135 nm), calculated as √(width×depth). The width of link channel is either smaller (115 nm) or bigger (155 nm) than the upstream/downstream channel. C. Configuration space over which we solve the Fokker Planck equation. The y dimension (upstream) is omitted because the preserved total contour length of DNA reduces the number of dimensions and sets the diagonal boundary. The right boundary represents the finite length of the link. D. Probability of DNA entering wide or narrow link channel at the junction as function of flow speed. E. Time integrated image of multiple DNA transport events at flow speed ≈0.5 μm/sec. F. The numerical simulation of transport of multiple DNA molecules at flow speed ≈0.5 μm/sec. The transport modes are represented by the probability value shown in the scalebar. The vertical and the horizontal directions represent the undulating channels and link channels respectively.

In Figure 4B, we label the extended lengths of DNA segments in the link, upstream, and downstream channels as x, y, and z, respectively. The total effective potential for this configuration is the sum of the confinement free energy and the quasi-flow potential for each of the DNA segments and takes a parabolic form

$$\mathcal{H}=\left[x{\gamma }_x\right]+\left[y{\gamma }_y+\frac{1}{2}{\xi }_y{v}_{}{y}^2\right]+\left[z{\gamma }_z-2\frac{1}{2}{\xi }_z{v}_{}{z}^2\right]$$ (2)

Here, ${\gamma }_x,{\gamma }_y,$ and ${\gamma }_z$ are the confinement free energies per unit extended length of DNA in the link, upstream and downstream channels. ${\xi }_y$ and ${\xi }_z$ are friction coefficients per unit extended length of DNA chain in the upstream and downstream channels. v is mean flow speed of the buffer. In Equation 2, the first bracketed term $\left({\mathcal{H}}_x\right)$ is the total effective potential of DNA segment in the link channel, where the contribution comes only from the confinement free energy due to the absence of fluid flow. The second bracketed $\left({\mathcal{H}}_y\right)$ and the third bracketed $\left({\mathcal{H}}_z\right)$ terms are total effective potential of DNA segments in the upstream and the downstream channels. The flow potential in ${\mathcal{H}}_z$ is negative because the flow field pulls in the DNA into the downstream channel.

For the quantitative understanding of DNA transport within the junction model, we write the Langevin equations (Supporting Materials Equation S15, S16), and find the corresponding Fokker-Planck equation using the framework put by Lau and Lubensky.^[56] The Fokker-Planck equation for probability density function of DNA u(x,y,t) is

$$\frac{\partial u\left(x,y,t\right)}{\partial t}+\mathrm{\nabla }.\overrightarrow{J}=0$$ (3)

Here, $\overrightarrow{J}\left(x,y,t\right)=-\mathbb{D}\mathrm{\nabla }u-\beta \left(\mathbb{D}\mathrm{\nabla }\mathcal{H}\right)u$ is the probability flux, and $\beta ={(k_{B}T)}^{-1}.$ The configuration-dependent diffusivity tensor $\mathbb{D}$ is derived in the Supplementary Materials, and contains both an apparent friction of DNA relative to the walls, as well as the friction between two legs of a hernia. To match the boundaries of the integration domain with the experimental layout, we identify the boundary at $0\le x\le 1\text{\&}y=0$ with an exit into the downstream channel (Figure 4C). Because of the observed extension of DNA and in order to keep the model simple, we identify the boundaries at $1<x\le 3\text{\&}y=0\text{and}x=3\text{\&}y\ge 0$ with exit of the molecule through the link channel. We set the Dirichlet boundaries where DNA exits the junction. The constraint on contour length of DNA sets up the Neumann boundary along the diagonal of the simulation domain. For x = 0 boundary, where we experimentally do not observe a significant flux, we use a Neumann boundary. In principle, flux here would correspond to a cross-over to other molecular topologies, which we have excluded at the outset of the model. The initial configuration probability distribution is set as a 2D Gaussian distribution in the upstream channel.

The results of the numerical simulation show that the time evolution of the probability density function (PDF) of DNA at the junction with a wide link channel is biased toward the link channel for low flow speed ≲ 1.0 μm/sec (Figure S2A). However, at the junction with narrow link channel, the time evolution of the PDF is toward the downstream channel (Figure S2B). At relatively high flow speed (≳ 6.0 μm/sec), the time evolution of PDF of DNA is toward the downstream channel at the junction irrespective of the link channel size (Figure S2C-D). Qualitatively the above results agree well with the experimental data of DNA transport at a nanochannel junction.

To compare the numerical results with our stochastic experimental data, we calculate the probability of DNA entering a link channel at the junction for a range of flow speed. To minimize the impact of complex transport modes, we fabricated a simplified device with pairs of undulating channels that are connected by either wide or narrow link channels (Figure S3A). Experimentally, the probability is a ratio of the number of link channels DNA enters to the number of nanochannel junctions DNA encounters. Numerically, we define the probability as the time-integrated total probability flux out of the link channel boundaries

$$p=\underset{t}{\overset{}{{\displaystyle \int }}}\underset{\left\{link\right\}}{\overset{}{{\displaystyle \int }}}\overrightarrow{J}.\overrightarrow{n}dsdt$$ (4)

Here $\overrightarrow{n}$ is a unit normal away from the simulation boundaries. ds and dt are the length and time elements. The total time-integrated probability flux out of all boundaries is normalized to one.

The experimental and numerical results for the probability of DNA entering the link channel as function of flow speed are shown in Figure 4D. From the experiment, the probability of DNA passing through a wide link channel at the junction is very high (≈0.93) at flow speed ≈0.5 μm/sec, but it decreases quickly to ≈0.09 as the flow speed increases to ≈6.0 μm/sec. However, for a junction with narrow link channel, the probability is ≈0.05 regardless of the fluid flow speed. The numerical results agree closely with the experiments after optimization with the DNA-DNA friction ξ_m as the only free parameter (Supporting Materials)_. For a junction with a wide link channel, the numerical results have small deviation from the experimental data at the higher flow speed. We believe this is due to a few defective and clogged link channels during the fabrication or extended use of the device. The deviation could also be the result of complex molecular topologies of DNA at the junction that are not incorporated in our model. Overall, the above results show that our mean field dynamical model of DNA at the junction can largely explain the motion of DNA at a nanochannel junction.

For predicting the large-scale transport properties of DNA from a single junction model, we treat transport through each nanochannel junction as an independent event, as in a Markov process. Using the probabilities values (Equation 4), we write a set of transfer matrices for the transport along the flow axis, and simulate the transport of DNA in the metamaterial device. Figure 4F shows the transport of DNA at flow speed 0.5 μm/sec from simulation. Experimental visualization of transport of DNA through the device at flow speed ≈ 0.5 μm/sec is shown in Figure 4E, and bears great similarity to simulation results.

By using the transfer matrix, we can also track the probability that DNA exits the initial transport mode, and quantify the probability for a DNA molecule to take alternative paths. These events of mode-switching were also observed in our experimental results, where some DNA molecules “missed” the link channels that were in the path of the pure transport mode (Figure S4).

In summary, we showed three distinct transport modes of DNA in the metamaterial device, and how these modes could be manipulated through a change of driving pressure. For the quantitative description of the transport of DNA, we proposed a dynamical mean-field model of DNA at a single nanochannel junction, and expanded it to explain the global transport of DNA through the device. The results from our model agree well with the experimental stochastic data of DNA transport.

We believe that the concepts behind the model will be equally useful in studying the dynamics of DNA in various complex nanochannel geometries, which then can be used to optimize the nanofluidic device for the potential applications. We further believe this model could be further extended to understand the dynamics of other molecular topologies that exist at a nanochannel junction. However, presently some of the parameters used in the model are still not fully established, such as the scaling of friction coefficient,^{[1,8,13,57–60]} (details in Supporting Materials) mutual friction coefficient between two colocalized DNA chains in the nanochannel.^[61] Beyond this advance in understanding, we note that the specific design of parallel nanochannels connected by short, wider links can be directly used to transfer from liquid stream into another one. This is interesting because this the essential step in performing buffer exchanges that are central to protocols in molecular biology.