Dynamic motions of DNA molecules in an array of plasmonic traps

Abstract

DNA molecules can undergo dynamic motion by fluidic and plasmonic forces. The latter, very weak compared to the former, can alter the trajectory of DNA molecules from their original fluid direction. For analyzing the trajectory of DNA molecules, it is easier to view the force exerting plasmonic trap as a potential well. The potential well can be placed periodically forming a kinetically biased landscape. The motion of DNA molecules in this is the main subject of this study. In this paper, a scaled-up mock-up of DNA molecules and the unit-cell of the periodic potential landscape is fabricated by 3D printing and they are used to experimentally obtain the trajectory data. The data is then inputted into a computer simulation to check the trajectory of DNA molecules along the entire array of plasmonic traps. The novel analysis method provided in this paper can be used in the design of an array of plasmonic traps.

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Introduction

DNA molecules exhibit interesting morphologies. When stretched, they form a thread-like shape that can span up to a meter. However, they are capable of folding themselves into a complex shape that is small enough to fit inside any of our cells. For example, a 48 kbps (kilo base pairs) lambda DNA molecule has a hydrodynamic diameter of 1.4 μm but will stretch to 22 μm when pulled.¹ The aspect ratio of a DNA molecule is very high and the geometric diameter (the sectional diameter) of a DNA molecule is only 2 nm.² However, because it is highly charged, the effective diameter is 5 nm for 0.15 M concentration of NaCl.³

A length that is directly proportional to the number of base pairs is one of the most important features of a DNA molecule. The correct sorting of DNA molecules based on the number of base pairs is one of the most important processes in DNA sequencing. The most widely accepted process for DNA sorting is gel electrophoresis.⁴ This process requires many environmentally harmful electrochemical subprocesses and typically takes several hours.

Although not for DNA molecules, several attempts of using photonic forces for sorting spherical particles with different optical properties have been proposed.^5,6 Ladavac et al. have used periodic potential landscapes made by photonic forces to fractionate particles of different sizes.⁵ On the other hand, MacDonald et al. fractionated particles based on the different refractive indices of particles.⁶ Both methods were based on devising periodic regions of high laser intensities which pulled particles that encounter these attractive sites on their initial motion, therefore deflecting the original pathway imposed by the fluidic forces.

Handling of DNA molecules have been commonly done in bulk and only few works have been reported for the manipulation of a single DNA molecule.^7–12 Among the techniques, plasmonic tweezers are well suited for trapping nanoscale objects due to its unprecedented ability to confine field enhancements that are in the order of tens of nanometers.^13–22 There have been several reports on plasmonic trapping of nanoparticles such as DNA,^14–16 proteins,^17,18 polymer chain,¹⁹ and polystyrene nanoparticles^20–22 either in its natural state or after they were fluorescently labeled. Kim et al. have shown the trapping of a 4.7 kbps plasmid DNA and a 48 kbps lambda DNA in plasmonic tweezers with a single nanohole and differentiated them by measuring the increased scattering signal.¹⁴ Similarly, Kotnala et al. have used double-nanohole to trap a single 10 bps DNA.¹⁵ The double nanohole have even been used for trapping a single protein molecule called bovine serum albumin (BSA).¹⁷ Note that these molecules are not visible using microscopes and the trapping were verified by measuring the sudden increase in the scattering signal that resulted from the dielectric loading at the trap. On the other hand, Toshimitsu et al. have verified the tapping by monitoring the polymeric chain formation that is initiated when a plasmonic trap is formed at the double pyramids.¹⁹ Lastly, Kim et al. have constructed a side view system to visualize the fluorescence coming from a 300 nm polystyrene particle immobilized at the plasmonic trap, therefore, making it possible to characterize the maximal trapping force in a specially designed microfluidic channel.²⁰

Although DNAs have been trapped in plasmonic tweezers, questions remain about its dynamics. How do the DNA strands get picked up by the trap? How are they pulled towards the trap? And lastly, how do they settle and eventually escape from the trap? These questions are all very difficult to answer because most of our understanding of the dynamics in the optical trap is for rigid shapes such as spheres or rods.

Traditionally in engineering, one often makes a reduced-scale mockup to test drive the design. This is still prevalent today where well established computer-aided engineering simulation programs are available. For example, in aerospace, a reduced scale aircraft is tested in a wind-tunnel to assess the lift force.^23–25 We asked the exactly opposite question. What if we make a scaled up mock-up of a DNA and place it in a carefully crafted terrain that is manufactured in accordance to the energy landscape of the plasmonic trapping potential? Can we achieve similar results as we do in many engineering fields?

Electromagnetic field in the dielectric medium becomes strongly coupled with the electrons in the metallic surface and electromagnetic resonance can give rise to an exceptionally high field value at features of small radius of curvature.^26–28 When a hole is drilled on a metal film, the location of small radius of curvature is along the rim of the hole and the field is locally enhanced that typically amounts to several times of the incident field.^14,20 When there is a steep gradient in the electromagnetic field distribution, particles nearby with dielectric constants higher than the immersion medium will move in the direction of increasing field strength. The force that a particle experiences as a function of its location can be integrated to obtain a potential field.

Because energy is conserved, we can freely interchange between potential fields originating from different energy sources. For chain-like molecules, we will assume they are composed of finite number of spherical particles, and there exists no energy arising from interaction between the particles. In other words, the total potential energy is determined by summing individual potential energy from each particle.

According to the interchangeability between the optical and gravitational potential fields and the superposition assumption, we can simulate the motion of chain-like molecules by a mock-up composed of a chain and a terrain manufactured depending on the height of the optical potential field. When we want to simulate additional forces such as the fluidic force arising from the constant flow of the liquid in a fluidic channel, we can integrate the fluidic force to obtain the fluidic potential field and then convert the fluidic potential field to the gravitational potential field. In general, any forces, no matter what their origin is, can be universally converted to gravitational potential field.

Based on the understanding of the motion of a chain in a terrain corresponding to the universal gravitational potential field, we believe the motion of a chain-like molecule in a plasmonic trap can be understood. The benefits of this mock-up are as follows. Various designs of plasmonic trap can be simulated with a terrain manufactured according to the optical and fluidic potential field. We can avoid the time-consuming and expensive investment needed to manufacture the plasmonic patterns. More importantly, as far as we know, there has been no understanding about the motion of a chain-like molecule in a plasmonic trap. Only a single bead-shaped particle has been investigated.²⁹ We believe the work described in this paper expands our understanding of plasmonic trapping beyond a simple spherical object to a flexible chain-like molecule, a challenge that has not been attempted.

Experimental

Chain link fabrication

The mechanical dynamics of chain-like molecules such as DNA have traditionally been modeled using freely-jointed models.³⁰ A unique value called persistence length determines many mechanical properties such as dynamical motions.³¹ The Kramers freely jointed bead-rod chain model has shown quantitative agreement with experiments in single molecule flow studies of DNA even though it does not correctly reproduce bending forces or entropic spring force.³² Using the Kramers chain model, a single DNA is modeled with a series of chain-units connected together. The thickness and size of a single chain-unit was based on the precision of the 3D printer and the durability of it to withstand a free fall from the height of 1 meter. The chain assembled by connecting the chain-units underwent random motions for each free fall, and the persistent length was measured each time and at the end the average end to end distance was calculated.

For fabricating the chain, we used a 3D printer (Formlabs, Form 1+). The chain is composed of a repeated chain-unit that is used to represent the chain link with a particular length that is freely rotating with respect to its immediate neighboring link. We have designed this unit using a commercial CAD software and the result is shown in Fig. 1a. When manufactured at the smallest curable spot size (300 μm), we found the links would not rotate freely at the two ends and would tear away when the chain underwent free motions. For thicker cured line widths, the chain would not show free frictionless rotations. Therefore we settled on 0.5 mm for the cured line width. After settling with the line width, the chain-units were fabricated with various aspect ratios from 1 : 2 up to 1 : 5. They were then placed under a free fall test. The test has shown that the most endurable aspect ratio was 1 : 2. Aspect ratios of 1 : 3 and above have shown to fracture as the result of impact to the ground. Some of the different aspect ratios we have tried are shown in Fig. 1b–d. It was found that Fig. 1c suits the durability test. Two unit parts were connected by slightly cutting the mating loops and carefully deforming each loop prior to crossing the loops together. The cut was mended with an epoxy. This process was repeated for 155 unit parts; the number being limited only by the effort we have put onto it.

Fig. 1 Design and 3D printed chain-unit. (a) Design of a chain-unit. (b) Half sized chain-unit that was damaged in a free-fall test. (c) Used chain-unit. (d) Double sized chain-unit. The cut at the upper right corner was purposely made with a cutter so that it would form a ring pairing to another chain-unit (not shown).

Persistence length based on freely jointed chain model

To experiment the configuration space of the mock-up chain, we have chosen to let it fall freely at 1 meter from the table surface. 1 meter was assumed to be enough to provide a random motion of the mock-up chain. A rubber plate was placed on top of a table for damping the impact during collision. The particular configuration of the mock-up chain after resting on the table surface was assumed to be a random configuration and the end-to-end distance was measured to calculate the persistent length as shown in eqn (1):where s is the end to end distance, n is the number of links and p is the persistent length of the link.³³ To have a statistically meaningful data, the number of experiments were performed that provides 90% confidence level assuming the experiment follows a normal distribution. Eqn (2) gives the required number of experiments:where N is the required number of experiments, Z_a/2 is the reliability, E is the permissible error.³⁴ For 10% admissible error, Z_a/2 and E are 1.645 and 0.1, respectively. From eqn (2), we can see that the number of experiments should be at least 68. We have done 70 number of experiments and calculated the persistent length based on these experiments. The calculated persistence length was 19.88 mm.

Scaled-up ratio between the mock-up and the DNA molecule

The persistent length of a DNA molecule is 53 nm and roughly consists of 150 bps.³⁵ The length of our mock-up chain was measured as 790 mm and we have previously calculated the persistence length to be 19.88 mm. If we assume our mock-up chain represents a scaled-up version of the DNA molecule, the ratio between the chain and the DNA molecule is 375 000 to 1.

Thus, this mock-up represents a DNA molecule of 36 persistent lengths and a little math gives 5.4 kbps is the length of the chain.

Prediction of dynamics of DNA near the plasmonic trap and its modeling

When a laser is incident on this surface, high field enhancement will occur at the rim of the hole. This is called surface plasmon resonance. The localized field enhancement applies a pulling force for nearby particles of higher dielectric constant than the surrounding medium.³⁶ When a DNA passes near a plasmonic nanohole, it will experience complex motions as shown in Fig. 2. DNA molecule will be picked up by the plasmonic force that will be directed towards the rim of the nanohole. At the same time, it will experience a fluid force in the direction of the downstream. The fluid force designed to be larger than the trapping force will eventually send the DNA past the plasmonic trap where it will be solely influenced further by only the fluid force. If we obliquely place two nanoholes at close proximity defined as a unit-cell, the DNA will experience the complex motion. To explain in detail, the DNA escaped from the first plasmonic trap will reenter the second trap. These pairing of two plasmonic traps works as a deflector. We can extend these unit-cells horizontally, vertically and also diagonally. These repeated unit-cells form a very unique pattern that we call the oblique pattern because it is slightly slanted forming an angle in both of the horizontal and vertical axes. This oblique pattern will continuously drive the DNA at certain deflection angle.

Fig. 2 Dynamic motions of a DNA near the plasmonic nanohole. The route of the DNA is deflected by the gradient force.

The geometry such as the diameter of the nanohole and the length of the chain molecule determines the maximal applicable trapping force. It can be understood that stronger force can apply larger angle of deflection because the oblique angle of the pattern demands certain minimal amount of trapping force. If the oblique angle of a particularly designed pattern is greater than deflection angle formed by the maximal amount of trapping force that the plasmonic hole can apply, the chain molecule of interest will derail from the path and move downwards in the direction of the fluid flow rather than following the designed oblique angle. In other words, we can assume that there will be a critical length of the chain molecule. If the length is longer than this critical length, the chain molecule will derail and fall out from the oblique angle. On the other hand, when the length is shorter, it will continue and follow the oblique angle.

F(r) = −∇U(r) (3)

The energy field can be obtained by spatially integrating the force field through eqn (3).³⁷ Notice the force and the energy field are all due to the polarization of the dielectric object which is a DNA in our case. When it comes to energy, it does not really matter what was the source of the energy. In other words, we can freely substitute the optical energy field to an energy other than the original nature of the energy as long as the energy satisfies eqn (3). This is exactly what we have done. We substituted the optical energy to a gravitational potential energy. Subsequently, by using eqn (3) and applying the gradient operation to the energy, we obtained the gravitational forces. The significance of this conversion is that we can devise a terrain with the height field exactly matching the energy obtained from the optical (polarization) force field. A mass placed on this terrain will follow a motion trajectory based on the force field through eqn (3) with the following assumptions. It should be noted that particles can undergo rotational motions through torques, however, we assume that there is no rotation motions. Thus, particles and chains placed on the terrain will purely slide. Furthermore, sliding incurs dissipative frictional (damping) forces. Optical trapping happens in a fluidic environment and the motions triggered by the optical forces will quickly dampen by the viscous drag from the immersed medium. Similarly, the motion of the particles and the chains on the terrain will undergo frictional forces that quickly dampens any motions coming from the gravitation forces. We assumed the frictional damping forces can substitute the role of the viscous drag forces.

To calculate the height field in theory, we notice it is dependent on the initial configuration of the chain molecule which can have several hundred degrees of freedom. For example, the angle between each joints between two chains consumes one degree of freedom. We can quickly see the complexity is overwhelmingly high to the point of being impractical. Thus, to bring down the complexity to a manageable level, but still providing meaningful results, we consider following important assumptions. The height field is based on computing a single particle whose size is infinitesimally small. In other words, it is a Rayleigh particle. And rather than physically computing this height field numerically, we assume this height field is very thin and forms a loop around the rim of the plasmonic nanohole. This height field representing the potential is simplified as a ditch that is placed right on the periphery of the plasmonic nanohole. The shape looks like a lower half side of a torus. The diameter of the ditch is 2 cm. It then relates to 53.3 nm when scaled-down by the scale factor. This diameter is designed to be small and effects the magnitude of an optical force field that the DNA experiences. The optical force is experimentally obtained from the scaled-up experiment of the trajectory of the chain through this potential well. The diameter of the major axis of the toroidal ditch is 15 cm that corresponds to 400 nm in nanoscale. We have fabricated this channel using a 3D printer (Union-tek, RS6000) with the shape modeling done using a commercial CAD software.

Experimental setup

Fig. 3a illustrates the experimental setup. This setup consists of two toroidal potential well which represents the plasmonic potential landscape and it is shown in Fig. 3b. The choices of variables in the experiments are d, θ, Φ. Firstly, d is the distance between two consecutive plasmonic nanoholes. Notice this distance is also a critical factor and determines whether the chain molecule will be on track and follow the oblique angle. There is a maximal distance d where further separation will also result in pushing the chain molecule off the track from the path designated by the oblique angle. Next, θ is the oblique angle that the two consecutive toroidal potential well is inclined to the direction of the fluid flow. When this angle becomes larger than the critical angle, chain will derail from the oblique angle. Lastly, Φ is the angle between the potential landscape and the experimental bed. Increasing this angle will apply stronger gravitation force to the chain. Φ relates to the fluid flow velocity in the real space. In the real space, the fluid velocity is controlled by an external pump. This angle determines the falling speed of the chain and is similar to the recreational ski slope angle.

Fig. 3 Experimental setup. d, θ, Φ are the controllable values of experimental bed. (a) Overall experimental setup. (b) Experimental bed. Chains flow downwards and they are affected by toroidal potential well.

Typically, the number of 400 nm nanoholes that are patterned in an area of 1 × 1 mm² pattern array is in the order of ten thousands. And it becomes impractical to 3D print this amount of holes in the scaled-up model. Our idea is to 3D print only a single unit-cell as shown in Fig. 3 and obtain the experimental motion of a chain that undergoes the designed potential well. The experimental results are then input to a computer simulation program that periodically expands the experimental results thereby calculating the result coming from an array of unit-cells. The calculated result will enable us to understand the motion of DNA molecules in an array of plasmonic traps.

The trajectory of a chain

Fig. 4 illustrates the trajectory of a chain placed at the top edge of the experimental board that is inclined to the ground to impart gravitational forces to the chain. The chain would fall straight downwards if there were no inscribed potential well. However, due to the existence of the dual potential wells, the chain deviated from the straight path. The top and bottom edges will be called inlet and outlet, respectively, from here on. Fig. 4a shows the center points of the chain calculated at 40 millisecond intervals. We can see that the chain assumes initial straight down fall before hitting the potential wells, when it deflects due to being pulled by the potential wells. The path of the chain is determined by the horizontal location from the inlet. Fig. 4 shows a typical example from starting about one third from the left of the top edge. Obviously, starting at different horizontal locations will result in different pathways due to the change in the forces that the chain experiences from the potential wells. The pathway is also dependent on the shape of the chain which is random by nature. Even if the chain initiates from the same horizontal location, the trajectory can be completely different. Thus, to obtain a meaningful data, the experiments should be undertaken for multiple number of times from the same horizontal location. The repetition number was seventy times, the same number we have used to experimentally obtain the persistence length of a DNA molecule.

Fig. 4 Tracing of a chain trajectory. Dots represent the tracing points of a chain sampled at 40 ms interval. Position of a chain at the time, (a) 40 ms, (b) 160 ms, (c) 200 ms, (d) 280 ms, (e) 360 ms and (f) 440 ms.

The time lapsed trace of the centroids of the chain molecule is influenced by the gradient force from the toroidal potential well. The first four trace points from the top in Fig. 4a are under the influence of the upper potential well. It might appear that the chain molecule representing the fourth trace point marked as a triangle is outside the influence of the upper potential well. However, this is not true. As shown in Fig. 4b at the time 160 ms, we can verify that the tail of the chain molecule is still influenced by the upper potential well although most of the chain molecule is outside the upper potential well. Thus, the motion of the chain molecule is influenced by the gradient force at this location. The immediately following three star shaped trace points (fifth, sixth and seventh) represent the centroids of the chain molecule that are completely outside the influence of the upper potential well. Because the chain molecules are not influenced by the gradient force, the trajectory of them has a straight motion.

In summary, the trajectory of the chain molecule shown in Fig. 4a can be divided into three sections. They are the first section influenced by the upper potential well, next section showing a straight motion, and the last section that is influenced by the bottom potential well. In other words, all trace points except the star shaped trace points are under the influence of the gradient force and shows a curved motion. The motion of particles in the order of micrometers generally exhibit overdamped condition. In other words, the inertial force is much smaller than the viscous force. Above experiment have shown this characteristic and the chain molecule assumes a straight motion when there are no external forces applied. In the section “Calculation of the gradient force and laser intensity requirement for the physical realization of the experiment”, we provide some quantitative analysis to support the overdamped motion.

Accumulation pattern at the outlet

Fig. 5 illustrates the count at the corresponding horizontal location from the left at the exiting outlet from the downward fall from the top inlet locations. We have normalized the counts to treat the outcome as a probability distribution. As we have noted earlier, the distribution at the outlet strongly depends on the inlet location. If the chain underwent planar potential, it will move in a straight line making the outlet location to be forming a Gaussian distribution centered at the same distance from the left. However, when there are potential wells on its course, it will deviate and form a certain distribution at the outlet. Fig. 5a shows some typical initiating locations marked A, B and C. For A and C, the chain will never meet the potential well and the resulting exiting locations will distribute as shown in Fig. 5b, A and C. However, when it starts at the location B in Fig. 5a, it will be influenced by the potential well and exit at locations when plotted would be the central twin peaks, B, in Fig. 5b. Notice the rather broadened shape of the distribution B. Above mentioned examples illustrate the dependent nature of the exiting distributions to the inlet locations.

Fig. 5 Accumulation results at the outlet. (a) A chain is initiated from A, B, C locations. A chain dropped from A and C are not affected by the potential wells. Opposite is true for B. (b) Theoretical accumulations at the outlet bin. A and C show a Gaussian distribution centered from the vertically aligned locations from the drop locations. B shows broadened and biased accumulation results. (c) Sampled data collected from finitely size collecting bins at the outlet.

It would be practically impossible to test all possible horizontal locations of the imitating chains. Thus, to reduce the test to a manageable number, we specify regular intervals both for the starting inlet locations and the measured exiting outlet locations. When counting the exit locations, we consider all locations within the interval distance to be exiting at the center of the bin. The length of the inlet and outlet were 80 cm and we have chosen 2 cm as the interval distance. The resulting graph using the outlet bins is shown in Fig. 5c.

Active part of the unit-cell

Fig. 6 illustrates the accumulation results at the outlet when the chains are loaded from certain distance from the left edge of the unit-cell. The chains are loaded from the inlet for finite number of trials and also at finite number of drop sites. These drop locations are also identified as bins and the center location of the bin defines the exact drop location. In this real experiment, we have repeatedly performed the loading experiments from each inlet bin for 70 times and the average location of the outlet bin is calculated and plotted as small empty circles. Similar to the average of the plotted functions A and C from Fig. 5a, notice for certain inlet bins, the average accumulated bin location at the outlet is the same. This means the chain went straight down. For these paired inlet and outlet bins, the potential well had no effects on the trajectory of the chain. On the other hand, we also have non-corresponding inlet and outlet pairs that are in the middle of the plot marked by the horizontal double arrows and labeled as effected bins. Effected bins at the outlet are a subset of the total outlet bins that are influenced by the potential well and they are the key driving force for biasing the trajectory of the chain into a designed direction. In Fig. 6, the effected bins include the bins from 13th to 28th and this horizontal span is the active part in a unit-cell. We call this part as the active part of the unit-cell and it depends on the experimental conditions such as d and θ as shown in Fig. 3.

Fig. 6 The correspondences between the sampled inlet and outlet bins indices. Effected bins are those index locations where the inlet and outlet indices are different because of the effect of the potential well.

Biased motion of the chain

Fig. 7a and c illustrate the chain launched from a horizontal location such that it is tangent to the left side of the upper potential well (13th bin) and the right side of the lower potential well (28th bin). The resulting cumulative exiting locations at the outlet are also drawn as a distribution function in Fig. 7b and d where the vertical axis denotes the percentage of the chain that exited at the corresponding bin shown on the horizontal axis. Integrating the distribution function gives 1.0 which denotes the total number of trials done at an identical location at the inlet. As previously dictated by the statistical confidence level, the total number of trials were 70. The hatched areas P1 and P2 shown in Fig. 7b and d denote the exiting chains that cross the active part of the unit-cell (shown as the dotted rectangle in Fig. 7a and c). The area under P1 denotes the chains that cross the left boundary of the unit-cell and oppositely, P2 is one that cross the right boundary. Note the two areas P1 and P2 are not generally equal. This means the chains will be migrating more dominantly to the direction that shows greater area. This is an important aspect that plays a vital role in accumulating the chain into the intended direction. Next, we will explain how the active part of the unit-cell is copied horizontally and then vertically to model the entire array of potential wells.

Fig. 7 Probability distribution at the end of the domain depending on the location of the drop site from the left edge. (a and c) The launched chains at the 13th and 28th bin, respectively. (b and d) The cumulative exiting locations at the outlet as a distribution function dependence on the location of the launched chains.

Extending to an array of potential wells

Fig. 8 illustrates the array of the active part of the unit-cells that are repeatedly placed horizontally and vertically. It models a physical plasmonic chip that can be manufactured. Basically, the trajectory of a chain in the array can be deduced from a computer simulation program that utilizes the experimentally obtained outlet distributions at the indexed inlet bins. The simulation program assembles the adjacent unit-cells by feeding the exiting chains to the inlets of the receiving unit-cell. In Fig. 8, the top boundary of the chip that is formed by the horizontally placed unit-cells at the top is referred to as boundary I. Subsequently, the top boundary of the second row of horizontally placed unit-cells is called S₁. And the following boundaries are called S_i where, i runs from 1 to n. Because we want the potential wells to smoothly transition from the top layer of the unit-cells to the bottom layer of the unit-cells, the bottom layer of the unit-cells need to be offset by a marginal distance from the left boundary of the chip. For example, the left edge of the topmost horizontal stripe of the unit-cells and the corresponding edge of the immediately adjacent second stripe of the unit-cells are not vertically aligned as shown in Fig. 8. Rather, the second stripe starts by slightly offsetting by the horizontal distance that is equal to the width of the hatched rectangle. This margin gets wider as we move downwards. When the margin gets wider than the width of the unit-cell, a new unit-cell is placed left adjusted to the horizontal array of unit-cells thus always restricting the width of the margin to be smaller than the width of a unit-cell. An opposite problem happens on the right side of the chip. Notice because the stripe of the horizontal unit-cells are shifted to the right, there will be not enough space to place the unit-cell to the right. These spaces are filled with planar potential block so that the partial unit-cells are not used on the right side. We also need to consider an appropriate boundary conditions at the left and right boundary of the chip. In the patterned chip, these edges pertain to the side walls of a microfluidic channel and would block the DNA molecules from going further passed the side walls. We assume the DNA molecule would collide the side walls inelastically. This means the kinetic energy will be partially absorbed by the walls and the DNA molecules will bounce back with an energy less than that prior to the collision. The amount that it will bounce back will be depending on the materials of the side wall. Typically, PDMS (polydimethylsiloxane) is used for the side wall. With above mentioned strategy implemented to the periodically expanded unit-cells in the horizontal and vertical directions with proper boundary conditions, the accumulation of the DNA molecules at the bottom of the plasmonic chip can be numerically obtained.

Fig. 8 Plasmonic chip. Width is consisted with 30 unit-cells and plasmonic pattern array chip is made by repeated unit-cell. And both lateral side of plasmonic chip is blocked by fluid channel. Thus, DNAs cannot pass over edge of plasmonic pattern array chip.

Scale factors that govern the magnitude of the forces in the macro and nanoscale

In order for our scaled-up experiment to substitute for the nanoscale physics, the scaling effect needs to be accounted for. We first examine how the scaling affects the forces in the macroscale. Eqn (4) illustrates the total acting force as the sum of four constitutive forces that are gravitational, potential, buoyant and frictional forces as shown in Fig. 9.

Fig. 9 The movement of a spherical particle that approximates a chain and forces that affect it on the experimental bed. (a) Overall chain movements at the toroidal potential well. (b) Enlarged view of the chain movement. (c) Forces that affect the chains. (d) Sectional force diagram.

Fig. 9a illustrates the motion of a spherical particle that follows the potential well. Fig. 9b is an enlarged view of the inset in Fig. 9a. Notice we have substituted the chain with a sphere under the assumption that the superposition rules do apply as previously explained. Each force term in eqn (4) is illustrated in Fig. 9c and d with an arrow showing the direction of the forces. The potential force denotes the plasmonic force that pulls the chain towards the bottom of the toroidal ditch. The direction of the force will be slightly slanted from the vector towards the bottom of the toroidal ditch because the plate that hosts the potential well is inclined at an angle to the floor. The buoyant force (F_buoyancy) is ρVg (ρ: the density of air, V: the volume of the chain, g: the gravitational acceleration) and we get 3.32 × 10⁻⁷ N based on putting relevant physical values. The gravitational force (F_gravity) on the chain from mg (m: the mass of the chain, g: the gravitational acceleration), gives 9.75 × 10⁻² N which is about 16 000 times larger than the buoyant force. The kinematic frictional force (F_friction) can be obtained by multiplying kinematic frictional constant to the gravitational force. This force is determined by the contacting surface roughness and it is not effected by the size of the sphere. The potential force (F_potential) obtained by multiplying the gravitational force by sin α. The kinematic frictional and potential forces should be considered because these forces are much larger than the buoyant forces as well as the gravitational forces. Thus, we come up with a simplified eqn (5), given below.

Notice all of the forces given in eqn (5) is proportional to the volume of the sphere as long as the density remains the same. Thus, if the diameter of the sphere is defined as L, the total force action on the sphere is proportional to L³.

We can also conduct similar analysis for the nanoscale physics. In short, the forces acting in the nanoscale can be written as eqn (6) and each force is shown in Fig. 10.

Fig. 10 Force diagram on the DNA. Field is enhanced by a plasmonic structure and the resulting gradient force affects the DNA. F_fluid represents the fluid force experienced by the DNA and F_drag represents the drag force experienced by the motion of the DNA.

The first term, the fluid force (F_fluid) can be computed by the drag force equation, 6πηrv (η: the viscosity of the medium, r: the radius of the sphere, v: the fluid velocity). It can be computed as 6.04 × 10⁻¹⁵ N. The fourth term, the drag force (F_drag) can be similarly found by substituting v with v′ which is the velocity of the sphere. The buoyant (F_buoyancy′) and gravitational (F_gravity′) forces are ρVg and mg. These can be computed as 5.76 × 10⁻²⁰ and 5.83 × 10⁻²⁰ N, respectively. These forces are five orders smaller than the fluid force and can be neglected. Finally, the gradient force (F_gradient) can be written as eqn (7), assuming the DNA molecule can be considered as a sphere.

here, n_md is the refractive index of the medium; c is the speed of light; m is the relative refractive index of the DNA to the medium; I is the intensity of the laser. By neglecting the buoyant and the gravitational forces, we can get a simplified representation of the acting force in the nanoscale as shown in eqn (8).

The fluid, gradient and drag force in eqn (8) can be related to the gravitational, potential and frictional force shown in eqn (5). Thus, the one to one mapping between the constitutive forces in macro and nanoscale can be made.

The fluid and the drag forces are proportional to the radius of the sphere r, and the velocity of the fluid v or the drag v′. These forces are thus proportional to L². The gradient force given in eqn (7) is determined by physical constants such as refractive index, relative refractive index and the speed of light that is invariant to the scaling effect. Moreover, the laser intensity is the source power divided the illuminated area and has no relation to the scaling effect. The only scale factor affecting the gradient force is r³ thus we can conclude that it is proportional to L³. The dominating force that affects the nanoscale governing equation in terms of the scaling effect is the gradient force that is proportional to L³. Since both of the governing equations in macro and nanoscale are proportional to L³, we can justify the fact that we can scale up the nanoscale physics to a macroscale mock-up.

Calculation of the gradient force and laser intensity requirement for the physical realization of the experiment

The fabricated potential well imparts a force to the chain in contact. We concentrated on the force that the potential well imposes on the chain. In this circumstance, the chain will enter the ditch of the toroidal potential and momentarily travel along the circumference of the circular trajectory. Since all other forces such as the gravitational force nor the frictional force cannot impose a circular motion of the sphere, this circular motion is solely coming from the potential force. The radial acceleration of the chain can be computed from the three consecutive center points of the chain in the video analysis. This radial acceleration multiplied by the mass of the chain gives the radial force. Note that this force is coming solely from the potential well and can be considered as the potential force. Finally, the gradient force in nanoscale can be obtained by reducing the potential force by the scale factor L³.

The gradient force can be used to deduce the required laser intensity. First, we need to calculate the optical force that a plasmonic trap exerts on a particle using the finite-difference time-domain (FDTD) method (Fig. S1†). Since the force is directly proportional to the laser intensity, the ratio between the computed optical force and the experimentally obtained gradient force can be used to find the right amount of laser intensity. The potential force is 7.51 × 10⁻² N calculated by multiplying the radial acceleration (7.7 m s⁻²) obtained from the video analysis and the mass of the chain (9.75 g). Scaling this down by L³ gives us 1.42 × 10⁻¹⁸ N. The laser intensity that is needed to impose this amount of the gradient force was found to be 6.61 μW μm⁻².

The inertial force can be computed by multiplying the acceleration of the chain molecule by its mass. The acceleration was obtained using a video analysis and scaled down by the scale factor (L). The inertial force components in the x and y axes were 7.52 × 10⁻¹⁹ N and 1.2 × 10⁻¹⁸ N, respectively. In contrast, the viscous force components in the x and y axes were 2.15 × 10⁻¹⁵ N and 4.69 × 10⁻¹⁵ N, respectively. The ratio of the viscous force components to the inertial force components in the x and y axes were 2860 and 3910, respectively. Because the viscous force is three orders greater than the inertial force, we can conclude that the chain molecule undergoes overdamped motion.

Results and discussion

When the chains are injected to the stream within the active part of the unit-cell shown in Fig. 6, they will accumulate in a pattern dependent on the horizontal location at the inlet. Some exemplary accumulation patterns for the entry locations at the two far ends are shown in Fig. 7b and d. Notice the two accumulation patterns are not symmetric and the difference of the areas that are outside the active part of the unit-cell determines how biased the trajectory of the chains will be at the outlet. For the correct calculation of the trajectory, we need to consider all accumulation patterns that originates at the inlet of the active part of the unit-cell. The calculation has shown that the summed movement of the chains within a unit-cell were biased to the right. This biased motion of the chains become stronger when we extend the unit-cell horizontally as well as vertically to form a plasmonic chip as shown in Fig. 8. This figure shows that we have 30 unit-cells in the horizontal direction. This horizontal row is repeated vertically and we can easily understand the biased motion of the chains will repeat; hence the chains will move further to the right. The design of the plasmonic chip enables better accumulation of the DNA to the side as the number of stripes in the vertical direction increases. This is an obvious result because each horizontal layer moves the whole population of chains further to the side. Fig. 8 illustrates this for a specifically designed plasmonic chip that only differs by the number of layers. The H denotes the length of the plasmonic chip.

The total accumulation pattern in a plasmonic chip is directly affected by the accumulation pattern which is related with H, d, θ, v in a unit-cell. These values represent the length of the plasmonic chip, the center to center distance between two consecutive nanoholes, the angle formed between the segment that joins these two centers and the vertical axis and lastly, the fluid velocity, respectively. We have performed an extensive experiment varying these values to understand how the values affect to the accumulation in a plasmonic chip.

We first examined the effect of the length of the plasmonic chip to the accumulation pattern. Fig. 11 illustrates plots that differ only by the chip length. The values of d, v, θ were fixed and the chip lengths were tested at 100, 500, 1000, 1500 μm. Corresponding results are shown from Fig. 11a–d. We can confirm that the chain tends to accumulate at the right hand corner as the length of H increases.

Fig. 11 Accumulation pattern at the chip bottom as a function of chip length. Accumulation increases as H increases. Each H is (a) 100 μm, (b) 500 μm, (c) 1000 μm, (d) 1500 μm. Here, the conditions of d, v, θ, I are same as 460 nm, 2.03 μm s⁻¹, 10°, 6.61 μW μm⁻², respectively.

The effect of d, θ, v on the accumulation pattern at the chip end were tested and the details are shown in Fig. S2–S4.† From these observations, we can conclude that the peak accumulation to the rightmost bin is a complex function of d, v and θ. Note above that these experimental results were conducted for one chain length. Generally, a chain of certain length under specific conditions gets accumulated to the rightmost bins by certain amounts. We define the accumulation efficiency as the inverse of the chip length H that is required to have 90% of the chains to be accumulated at the rightmost 10% bins. For the purpose of chain fractionations, it is imperative that the difference of the accumulation efficiency of two chains of different lengths ought to be large. In this way, we can obtain higher concentration for the tuned chain that has higher accumulation efficiency. Therefore, we have conducted experiments for series of the chains of 675 bps. In summary, the three kinds of d, three kinds of v, and four kinds of θ, giving rise to 36 combinations of conditions were used for two chains of different lengths and the results were assessed based on the difference in the accumulation efficiencies.

Fig. 12 shows the accumulation efficiency for 5.4 kbps chain and 675 bps under various conditions. We have normalized the accumulation efficiency by multiplying by 10 000. In physical terms, this normalization denotes the number of chains out of 10 000 that will migrate to the rightmost 10% bins. In Fig. 12, the accumulation efficiency decreases for longer d and peaks at certain θ.

Fig. 12 Accumulation efficiency of 5.4 kbps (a, c and e) and 675 bps (b, d and f) chain depending on the θ. Each condition has a peak point that represents the maximum accumulation efficiency. Graphs are separated in terms of d. Each d is (a and b) 460 nm, (c and d) 770 nm, (e and f) 1080 nm.

Fig. 13 shows plots that combines the results from Fig. 12. Fig. 13a–c show the accumulation efficiency for two lengths of chains as d increases. It is very interesting to see that 675 bps chain always show higher accumulation efficiency than the 5.4 kbps chain. This is due to the fact that shorter chains experience stronger gradient forces. Note this observation does not account for the additional forces such as the Brownian forces. For the purpose of fractionating chains of different lengths, it is important that one finds the conditions where the ratio in concentrations of the two chains at the collecting bins at the chip end is greatest. Imagine we load 675 bps chains at the chip beginning and count the number of chains at the 10% of bins to the right. We then redo this experiment with 5.4 kbps chains. Now the ratio of the two chains of different lengths defines the accumulation ratio. Higher accumulation ratio denotes the chip is better suited for the chain fractionation. In the fractionation process, these two chains will be mixed together forming a cocktail and loaded at the entry and collected at the down end.

Fig. 13 Comparison of efficiency of two different length chains dependence on d. Each d is (a) 460 nm, (b) 770 nm, (c) 1080 nm, respectively.

Fig. 14 plots the relation between the accumulation efficiency and ratio. One might think that the larger difference in the accumulation efficiency means that the accumulation ratio will also be larger. Surprisingly enough, this is not true as shown in Fig. 14. The fact that the difference in the accumulation efficiencies is larger does not necessarily mean that the accumulation ratio will also be larger. Thus, we can conclude that the accumulation efficiency difference cannot be used to find the most efficient fractionation conditions. Rather, one has to pay more attention to the accumulation ratio. The largest accumulation ratio can be found when H, d, v, θ are 700 μm, 770 nm, 2.03 μm s⁻¹, 20°, respectively. Fig. 15 visually shows the accumulation ratio for a 675 bps chain and a 5.4 kbps chain under prescribed conditions. Fig. 15a and b show that the concentration of 675 bps chain is three times than that of the 5.4 kbps chain. In order to obtain 99.99% pure 675 bps, the prescribed plasmonic chip can be used up to four times. The time it takes to pass one plasmonic chip takes 6 minutes making the total amount of time 24 minutes. Compare this to the gel electrophoresis where the process typically requires an hour to operate and the maximum purity can only be obtained up to 95%. In theory, the proposed method has a higher efficiency than this method.

Fig. 14 Relationship between accumulation efficiency and accumulation ratio of two different chains.

Fig. 15 Accumulation of 675 bps and 5.4 kbps DNA at the given condition that is the largest accumulation ratio.

Conclusions

In this work, we have studied the effect of an array of plasmonic traps on DNA molecules that is delivered through fluidic motion. This is simulated with a larger scale mock-up that was fabricated using 3D printing. The complex bending physics of DNA molecules have been realized using a 3D printed chain whose persistence length has been experimentally obtained. From the persistence length of the 3D printed chain, the scale factor (L) as divided by the persistence length of a DNA molecule has been calculated. This scale factor is then used to build a scaled-up mock-up of the potential landscape that is devised to apply scaled-up plasmonic forces. A video analysis was used to find the potential forces applied to the chain in the scale-up mock-up. Using a numerical electromagnetic solver, the laser intensity (6.61 μW μm⁻²) required to apply this amount of force scaled-down in the nanoscale was also computed.

Furthermore, we have investigated the accumulation efficiency of two chains having different lengths (675 bps and 5.4 kbps DNA) in terms of d, θ, v, that are the center to center distance between two consecutive nanoholes, the angle formed between the segment that joins these two centers and the vertical axis and lastly, the fluid velocity, respectively. We then applied this result to a sorting problem, the fractionation of two chains of different lengths. Through an extensive search of various combination of conditions, a procedure for obtaining high purity of certain chain lengths in a cocktail of two different lengths of chains were established. The proposed idea can in theory produce much higher purity of a DNA solution than the gel electrophoresis with less amount of time.

In summary, we have provided a novel method that combines the utilization of the experiments performed on a scaled-up mock-up and the computer simulation program based on periodically expanding the experimentally obtained results. Furthermore, we have presented how the forces in nanoscale can be obtained with scale factor. Finally, we have also presented how the array of plasmonic traps can be potentially used as a fractionation device. The method presented in this paper can be used as an invaluable sources to understand the effect of plasmonic traps on flexible chain-like molecules.

Acknowledgements

This work was supported by the ICT R&D program of MSIP/IITP [R0190-15-2040, Development of a contents configuration management system and a simulator for 3D printing using smart materials].

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Footnotes

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c6ra10414b

‡ These authors contributed equally to this work.

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