Fig 1.
Magnetic tweezers setup for perpendicular translation of bead chains.
(a): Schematic representation of the magnetic tweezers with a pair of cylinder magnets. The cylindrical magnets are magnetized in the axial direction, and they are paired up in an anti-parallel fashion. (b): Finite element simulation of the cylinder magnets pair using Finite Element Method Magnetics (FEMM 4.2). Cross section of the x, y-plane through the geometric center along z-axis is shown. Field density |B| decreases from pink to blue. N and S denote north and south poles respectively. Black lines represent computed magnetic field lines. Blue dot is a schematic field of view of the 50X objective, and blue dotted lines show zoom direction. Scale bar, 5 mm. (c): As viewed in the field of view, a 24-bead chain in the magnetic field of a pair of magnets, aligning with the field B (white arrow) along the y-axis, and translating towards the magnetic field gradient ∇B (black arrow). Scale bar, 10 μm.
Fig 2.
Magnetic tweezers setup for parallel translation of bead chains.
(a): Schematic representation of the magnetic tweezers with a single cone magnet, which is magnetized north at tip and south at base. (b): Finite element simulation of the cone magnet using Finite Element Method Magnetics (FEMM 4.2). Cross section of the x, y-plane through the geometric center along z-axis is shown. Field density |B| decreases from pink to blue. N and S denote north and south poles respectively. Black lines represent computed magnetic field lines. Blue dot is a schematic field of view of the 50X objective, and blue dotted lines show zoom direction. Scale bar, 5 mm. (c): A 21-bead chain in the magnetic field of a single cone magnet, aligning with the field B (white arrow) along the x-axis, and translating towards the magnetic field gradient ∇B (black arrow). The force applied to the particle is in the direction of the magnetic field gradient ∇B. Scale bar, 10 μm.
Fig 3.
Magnetic beads and bead-chains tracking with LabVIEW program.
(a): Displacement along the x-axis of the 24-bead chain from Fig 1c versus time plotted in red. Several single beads measured in the same experiment plotted in black. (b): Displacement along the x-axis of the 21-bead chain from Fig 2c versus time plotted in cyan. Several single beads measured in that same experiment plotted in black. The velocity of a given bead or bead-chain was calculated by dividing the displacement along x-axis by the time taken.
Fig 4.
Principle of drag coefficient calculations.
Schematic illustration of magnetic force and drag force acting on (a) single bead, (b) bead-chain moving perpendicular to its axial axis, and (c) bead-chain moving parallel to its axial axis. For a given bead-chain of length N, the drag coefficient is calculated using Eqs 9 and 11.
Table 1.
Correction factor γ∥ and γ⊥ from various theories for rod-like object.
Table 2.
Comparison of experimental and theoretical values ξN/ξ1 for bead-chains of length 2 to 30.
Fig 5.
Bead model and ellipsoid model theories and measurements.
(a): Schematic of the bead model and ellipsoid model theoretical frameworks representing rod-like objects. Bead model is a rigid array of identical touching beads each with diameter 2r; ellipsoid model is composed of major axis L/2 and minor axis r; Chain length in bead model in terms of number of beads N is equivalent to aspect ratio of ellipsoid L/(2r). (b): Experimental values for (red color dots) plotted along with experimental values for
(cyan color dots) versus chain length N. Theoretical curves are from Eqs 14 and 15 substituted with appropriate values of γ based on bead model and ellipsoid model as listed in Table 1, and HYDRO++ values in Table 2.
Fig 6.
Cylinder model theories and measurements.
(a): Schematic of the cylinder model theoretical framework representing rod-like objects. Cylinder model has diameter of 2r and length of L. Chain length in bead model in terms of number of beads N is equivalent to aspect ratio of ellipsoid and cylinder L/(2r). (b): The same experimental data from Fig 5b compared with theoretical curves with various γ from cylinder model as listed in Table 1. Black arrows point to theoretical curves obtained from Broersma [24].