High-throughput precise particle transport at single-particle resolution in a three-dimensional magnetic field for highly sensitive bio-detection

The applications of our tool can be divided into two main categories of “particle transport” and “analyte detection”. Hence, we divide this section into two sub-sections. In the first one, we mention the underlying equations governing particle transport. We explain why the circular pattern is problematic and how we overcome the challenge by applying a vertical bias field and using the drop-shape design. Then, we discuss the ability of the bent magnetic tracks in transporting magnetic particles of different sizes and come up with design rules. We provide both simulation and experimental results for various designs. The results in this sub-section are sufficient to design a circuit to accurately transport single-particles to desired spots on the chip.

Then, in the next sub-section, we present the experimental results for detecting analytes of interest, using the proposed tool. We show the ability of the device in detecting proteins and DNA fragments, captured by the ligand-functionalized magnetic beads. We compare the results obtained in a 3D magnetic field and show the improvement achieved compared to the results achieved when using the 2D field configuration.

Magnetophoretic circuit design

To appropriately design the magnetophoretic circuits, the relation between the magnetic potential energy and the magnetic force on a magnetic particle exposed to a magnetic field needs to be considered as stated in Eq. (1)⁵⁸:

where F and U are the magnetic force for transporting the particle from point r_i-1 to r_i and magnetic potential energy difference between the two points, respectively. The magnetic energy in Eq. (1) can be calculated as shown in Eq. (2)⁵⁸:

where μ₀, V_p, χ_p, χ_f, and H stand for vacuum magnetic permeability, particle volume, particle magnetic susceptibility, fluid magnetic susceptibility, and magnetic field intensity, respectively. Then, the velocity for a spherical magnetic particle in aqueous fluids (which is the case in our study), using an overdamped first-order motion equation, can be written as Eq. (3)⁵⁹:

where η_f and Rp represent the fluid viscosity and the particle radius. In order to predict the trajectory of the particle, it can be considered as a series of short straight paths (with laminar flow only) and a simple forward difference scheme, where (r_{i} = r_{i – 1} + v_{i – 1} Delta t) defines the particle position based on its previous position and velocity, can be used⁴⁶. The small friction (including rotational friction) forces can be neglected. Since our chips are covered with non-fouling layers, the chance of particles-surface bond formation is too low. Moreover, since the particles move around the magnets with a trajectory outside the magnetic pattern, we do not need to consider the dynamics of particles moving over the 100 nm thick magnetic thin film. Thus, based on Eq. (1), the magnetic particle exposed to a magnetic field gradient moves towards the area with minimum dipolar energy, and by studying energy distribution the particle trajectories can be predicted. We used Eq. (2) in COMSOL software to simulate the energy distribution (See “Materials and methods” section).

The locations of the dipolar energy minima in linear magnetizable systems are at points where the outward normal component of the magnetic pattern curvature is parallel to the external magnetic field. Thus, in the magnetic disks used in magnetophoretic circuits operating in 2D in-plane magnetic fields, two energy wells form on their opposite sides (i.e., the north and south poles). Magnetic particles and cells tend to move to these energy wells. In a rotating magnetic field, they periodically switch between the north and south poles and move along the magnetic tracks composed of connected magnetic disks⁴⁹.

The fundamental problem of manipulating single particles in magnetophoretic circuits operating in a 2D field comes from the micro-disk rotational symmetry in the in-plane rotating magnetic fields, where the dipoles cannot identify the north and south poles of the magnetic thin film patterns. Superimposing a vertical field to the in-plane rotating field (1) turns one of the energy wells (i.e., attractive poles) into an energy hill (i.e., repulsive pole) (See Fig. 1a,b for the simulation results, where blue and red area depict the regions with low and high magnetic energy) and (2) provides a repulsion force between the particles (See Fig. 1c,d). As shown in Fig. 1c (side view), in an in-plane magnetic field the opposite poles of the two adjacent magnetic particles meet and attract each other. However, as illustrated in Fig. 1d, when particles are biased, the force between them is repulsive. These attractive and repulsive forces between the two particles, as two magnetic point dipoles, can be described as Eq. (4)^60,61:

where (m_{i} = frac{{4pi R_{p}^{3} {upchi }_{p} H}}{3}) is the particle dipole moment and r stands for the distance between the two particles. In the cases in which the vertical fields are stronger than the in-plane field, the force in Eq. (4) becomes negative, depicting the repulsive force between the particles. The more complicated force calculation can be found elsewhere⁶². Although we did not carefully study whether the particles can overcome the downward gravity force and the downward magnetic force applied to the particle from the magnetic thin film, be levitated, and form vertical agglomerates, we did not see them being formed in our experiments.

In disk-based magnetophoretic circuits operating in a 2D in-plane magnetic field, as shown in Fig. 1e, the particles switch between the poles of adjacent magnetic disks. But, in the presence of the bias field and a rotating magnetic field, as illustrated in Fig. 1f, the particles move in a closed loop around a single disk and cannot move along the magnetic track (See the red dotted line in Fig. 1e,f for the particle trajectory in each case). As seen in Fig. 1f, since the pole at the adjacent disk shows an energy hill, the problem cannot be answered by adjusting parameters such as disk diameter or gap size. To overcome this challenge, we design magnetic patterns consisting of alternating sections of positive and negative curvatures so that particle transport in a 3D magnetic field is possible. The result is the drop-shape magnetic patterns, shown in Fig. 2, where particles move along the magnetic track⁵². In this figure, the numbers stand for the sequence of the particle positions when it moves along the magnetic track from one magnet (the “drop” section of the track) to another.

Figure 2 shows the transport mechanism in a 3D magnetic field on the drop-shape magnetic track which involves smooth transports along the sections with positive curvatures (e.g., from point 1 to point 3 in Fig. 2), followed by sudden transitions at sections with negative curvatures (e.g., from point 3 to point 4). We introduce β = d_P/G and γ = d_P/N as two dimensionless parameters, where d_P, G, and N stand for the particle diameter, the pattern gap, and the pattern neck size (See Fig. 2), respectively. In a straight magnetic track, similar to the one in Fig. 2, in our studies, we see appropriate particle transport for β > 0.14 and γ < 1.4.

To switch the particle transport direction between x- and y- directions, obtuse and acute bends are needed. We designed and tested several obtuse bends, an example of which is illustrated in Fig. 3. These patterns are designed by enlarging the gap between the magnet numbers 1 and 2 (and 2 and 3), which negatively affects β (i.e., they cannot transfer small particles). The goal here is to transport the particle from magnet 1 to magnet 3 through magnet 2. The magnetic energy distribution in Fig. 3a shows the blue areas at which particles initially stay (e.g., point p). By rotating the magnetic field in Fig. 3b, the particle at point p has two paths to choose to the blue areas of q₁ or q₂. By plotting the magnetic energy along lines pq₁ and pq₂ at heights of 2.5 µm and 10 µm in Fig. 3c,d, we realize that an energy barrier along the pq₂ path exists. Thus, the particle chooses the pq1 path and cannot move from magnet number 2 to magnet number 3. Our experimental results agree with these simulation predictions. In the inset of Fig. 3c, we show the experimental trajectories of two sample particles with blue and red dotted lines, respectively, circulating the magnets. The particle with the red trajectory in this figure has been placed on the magnet on the corner initially, and it cannot move to the next magnet on the track. The particle with the blue trajectory in this figure moves from one magnet to the next one in the area with an appropriate geometry (i.e., straight track). But, in the next step, due to an inappropriate β on the corner, it cannot move to the magnet there. Thus, this design is not appropriate for transporting magnetic particles. Figure 3e shows the experimentally measured particle velocity at various external field frequencies. Only rare particles (2 out of 15) randomly could jump forward at low frequencies, which cannot be considered proper particle transportation, resulting in larger error bars (standard deviation) compared to the results at higher frequencies. This phenomenon may be due to random particle parameters or defects. At high frequencies, we did not observe any jump between the magnets in the experiment, which may be because the particles do not have enough time to follow the energy wells that move fast (because of the large gap between the magnets in this geometry).

To overcome the challenge in the design shown in Fig. 3, we propose several other bend designs, in which we keep β and γ in the appropriate range. The first design in which the gap (G = 15 µm), and thus β, are kept very close to the one in the original straight tracks (i.e., the one shown in Fig. 2) is illustrated in Fig. 4a. In this design, the corner is replaced with many magnets each of which has a slight angle with respect to the main magnetic track. The energy simulation result for this design at the time of particle switching from one magnet to another is presented in Fig. 4b. The energy along paths pq₁ (backward path) and pq₂ (forward path) are shown in Fig. 4c,d, with the red and black curves, respectively. Figure 4c,d stand for the energies at the center of particles with diameters of 5 and 20 µm, respectively. The peaks in the red curves and the negative slopes in the black curves show that the particle moves forward along the magnetic track from one magnet to the next one. Our experiments show that the particle transport in this design is smooth (See red dotted lines in Fig. 4a for the experimental particle trajectories). We repeated these experiments at various frequencies (See Fig. 4e), and we found that the particles with different sizes (5–9.9) can move well at frequencies below ~ 0.6 Hz. But, the drawback of this design is the relatively large occupied space, due to the number of drop-shape magnets, this bend needs.

To enhance the bend design, we came up with the patterns shown in Figs. 5 and 6, where, as opposed to employing several drop-shape magnets, we use a single large magnet on the corner, which results in an appropriate β. The appropriate experimental trajectory shown in Fig. 5a is the result of the suitable energy distribution in Fig. 5b and is supported by the curves in Fig. 5c,d. Based on these curves the particle sees an energy barrier along the backward path (i.e., the red curves); however, it sees a negative energy slope along the forward path. Similarly, the energy distributions shown in Fig. 6a,b and the curves in Fig. 6c,d show that after reaching point p in Fig. 6a the particle moves to point q₂ in Fig. 6b. This behavior is seen in both particle sets with diameters of 5 and 20 µm. The experimental particle trajectories illustrated in Fig. 5a and the inset of Fig. 6c, confirm our theoretical findings. Our experimental results show that both designs work well in transporting particles of various sizes (5–9.9) at frequencies below ~ 0.6 Hz (See Figs. 5e, 6e).

Sometimes we need to transport the particles inside the bend (see Fig. 7). As shown in Fig. 7a,b the particle at point p in Fig. 7a reaches point q2 in Fig. 7b. The curves in Fig. 7c,d show that this design works fine for particle sizes in the range of 5 to 20 µm (i.e., the particles see an energy barrier in the backward path while they see an energy well in the forward path). A sample experimental particle trajectory on this bend design is shown in the inset of Fig. 7c. Again, we saw appropriate particle transport at frequencies below 0.6 Hz for the particle size range of 5–9.9 Hz.

The particle transport speed depends on the applied external magnetic field frequency. As shown in Figs. 4, 5, 6 and 7, the particles with the diameter range of 5–5.9 µm reach the average speed of ~ 25.2 µm/s at the frequency of 0.6, after which the particle cannot follow the external field (because of the big drag force compared to the driving magnetic force) and the velocity drops. The particles with the size range of 8–9.9 µm, because of higher magnetic susceptibilities can move at the frequency of 0.8 Hz and reach the average speed of ~ 32.4 µm/s. Since we need complete stable conditions for our particle transports and sensing tests, we chose the frequency of 0.2 Hz, with an average speed of ~ 9 µm/s, for our next experiments. We also tested the movement of small (e.g., 2.8 µm) and large 14–17.9 µm particles at 0.2 Hz and observed smooth transport on the proposed magnetophoretic circuits. Also, we ran the simulations in the range of 2–20 µm, but we chose to show the results of 5 µm and 10 µm beads, as two examples.

The combination of the introduced patterns provides the opportunity to design magnetophoretic circuits for the precise transportation of magnetic particles. Towards this goal, we designed a circuit to transport magnetic particles to spots on a chip with analytes of interest for detection purposes. Figure 8 illustrates a schematic of a sample circuit design.

Analyte detection

Here, we use our proposed magnetophoretic circuit as a tool to detect herpes simplex virus type 1 (HSV-1, or oral herpes) after transporting the bioparticles on-chip using the proposed magnetic circuit designs, without any need for amplification. The bend designs introduced in Figs. 6, 7 and magnetic beads with sizes of 2.8 µm (Dynabeads M-280), 6 µm (Bangs Laboratories 6 µm COMPEL) and 8 µm (Bangs Laboratories 6 µm COMPEL), resulting in (beta ,gamma approx 0.18); (beta ,gamma approx 0.4); and (beta ,gamma approx 0.53), respectively, were used. HSV is a DNA virus that belongs to the Herpesviridae family and affects more than 60% of the global population. Currently, culture-based or polymerase chain reaction-based methods are used for HSV diagnosis; however, more advanced sensitive and rapid HSV diagnostic tools are needed⁶³. Here, we use our proposed chip to run a single-molecule detection assay based on the reaction of the analyte of interest (e.g., HSV UL27 gene) in between two magnetic beads. Similar to recent works^55,56,57, in this method, we label the DNA fragments with biotin and digoxigenin oligonucleotide probes so that they can bind in between streptavidin and anti-digoxigenin antibody labeled magnetic beads, forming a magnetic bead pair/cluster (See Fig. 8), but, importantly, in a 3D magnetic field. To show the ability of the chip to detect multiple analytes simultaneously, we also use the chip to detect biotinylated bovine serum albumin (BSA), too. This detection is done via binding in between streptavidin-coated magnetic beads (See Fig. 8).

In a pure horizontal field (such as the fields in previous works), as shown in Fig. 1c, the magnetic beads attract each other and they may come in contact, even in the absence of the analyte of interest. This phenomenon may cause problems in demonstrating the real number of bead pairs linked with the analytes of interest. To overcome this challenge, in this work for the first time we use a vertical bias field, which intrinsically exists in our tool. In this magnetic field, in the absence of the analyte, the beads repel each other and do not come into contact (See Fig. 8a, where the red arrows depict the repulsion force between the beads). Thus, it lowers the chance of unwanted bead pair formation. But providing a link between the beads (See Fig. 8b) keeps them in contact. This link, as mentioned above, may be an analyte of interest such as a protein (See Fig. 8c) or a DNA fragment (See Fig. 8d).

Using the proposed magnetophoretic circuits in this work, we designed magnetic paths for transporting single magnetic beads to the spots (i.e., microchambers) with different analytes and various concentrations (See the schematic in Fig. 8e). After bead transportation, we first turn off the vertical bias field to let the beads meet, and then by turning it on, the beads with no link are separated, leaving only the detecting bead pairs (See Fig. 8h,j). We show that the number of bead pairs, formed in the microchambers from the loaded single beads, varies with respect to the available analyte concentration in each microchamber on the chip. Examples of a detected single bead and bead pair in a chamber with no analyte and high analyte concentrations, respectively, are shown in the insets of Fig. 8e as well as Fig. 8f,g where we use a simple image processing Matlab code to detect them. The experimental results based on the proposed method for the protein of interest and the DNA of interest are presented in Fig. 8j,m, respectively. These plots, as well as the box plots in Fig. 8k,l,n,o, show the close agreement between the results based on our proposed method (i.e., based on circuits operating in a 3D field) and FACS (Fluorescence-Activated Cell Sorting). But these plots depict that the device operating in the 2D field shows a larger number of pairs at low concentrations (< 10^–12) which makes analyte detection at these levels problematic (See flat curves in Fig. 8j,m at these concentrations). At high concentrations, the median and the error bars in the 2D field are larger than the ones based on the 3D field. These results confirm that our magnetophoretic circuits proposed in the current work operate as a more accurate biosensor.

Although it is out of the scope of the current work, detecting particle pairs after formation can also be transported. The trajectory of the particle pairs is similar to that of the single particles; however, since the drag force varies and, in addition to moving along the magnetic tracks, they experience a torque synced with the external magnetic field, the observed dynamic is more complicated.