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. 2010 Nov 15;108(10):104701.
doi: 10.1063/1.3510481. Epub 2010 Nov 18.

Quantitative modeling of forces in electromagnetic tweezers

Alex Bijamov, Fridon Shubitidze, Piercen M Oliver, Dmitri V Vezenov

Quantitative modeling of forces in electromagnetic tweezers

Alex Bijamov et al. J Appl Phys. .

Abstract

This paper discusses numerical simulations of the magnetic field produced by an electromagnet for generation of forces on superparamagnetic microspheres used in manipulation of single molecules or cells. Single molecule force spectroscopy based on magnetic tweezers can be used in applications that require parallel readout of biopolymer stretching or biomolecular binding. The magnetic tweezers exert forces on the surface-immobilized macromolecule by pulling a magnetic bead attached to the free end of the molecule in the direction of the field gradient. In a typical force spectroscopy experiment, the pulling forces can range between subpiconewton to tens of piconewtons. In order to effectively provide such forces, an understanding of the source of the magnetic field is required as the first step in the design of force spectroscopy systems. In this study, we use a numerical technique, the method of auxiliary sources, to investigate the influence of electromagnet geometry and material parameters of the magnetic core on the magnetic forces pulling the target beads in the area of interest. The close proximity of the area of interest to the magnet body results in deviations from intuitive relations between magnet size and pulling force, as well as in the force decay with distance. We discuss the benefits and drawbacks of various geometric modifications affecting the magnitude and spatial distribution of forces achievable with an electromagnet.

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Figures

Figure 1
Figure 1
(a) Typical geometry of the electromagnet that can be used in magnetic tweezers (not drawn to scale). (b) Geometry with the magnet showing conformal inner and outer surfaces for placement of auxiliary sources. EM field sources are placed on these auxiliary surfaces and aligned along the corresponding surface tangents (or normals). (c) Body of revolution magnet with surface tangents. The area of interest (hosting an array of biomolecules having length of ∼50–100 nm with magnetic beads of radius ∼1–3 μm attached to their ends) is 0.2–0.4 mm below the tip of the magnet.
Figure 2
Figure 2
(a) Magnetic B field distribution inside and outside of the magnet, (b) log10(B) distribution, (c) magnetic force distribution pulling the beads below the tip. Magnet tip radius=1.5 mm, tip length=10 mm, core radius=6 mm, core length=5 cm.
Figure 3
Figure 3
(a) Magnetic field and (b) magnetic force distribution along the magnet symmetry axis as a function of the distance from the tip with magnetic tip length varying from 5 to 15 mm. Magnet tip radius is fixed at 1.5 mm.
Figure 4
Figure 4
(a) Magnetic field and (b) magnetic force distribution along the magnetic core axis as a function of the distance from the tip with magnetic tip radius varying from 0.5 to 3 mm. Magnet tip length is fixed at 10 mm. (c) Pulling force, acting on magnetic beads positioned 0.5, 1.5, and 2.5 mm away from the magnet tip, as a function of the tip radius.
Figure 5
Figure 5
(a) Dependence of the radius of the magnet tip that provides highest pulling forces at a specific fixed location on the distance of this point from the magnet tip; (b) highest pulling force attainable by varying tip radius at a fixed distance from the sample for magnets with tip lengths of 5, 10, 15, and 20 mm. All other dimensions of the magnet core, coil parameters, and properties of magnetic beads are the same in all cases.
Figure 6
Figure 6
Ratio of the lateral pulling force to vertical force in the area of interest below the magnet tip, for magnetic tweezers with tip radii of (a) 0.5 mm, (b) 1.5 mm, and (c) 3 mm. Tip length=10 mm. Color map level step corresponds to 5% magnitude change in Fxy∕Fz.
Figure 7
Figure 7
(a) Magnetic field and (b) magnetic force distribution along the magnet symmetry axis as a function of the distance from the tip with magnet core length of 5, 10 and 15 cm. Magnet tip length is fixed at 10 mm, while the tip radius is 1.5 mm. Magnet core radius equals 6 mm. The coil is stretched to cover the entire length of the magnet core.
Figure 8
Figure 8
(a) magnetic field near the magnet tip, (b) magnetic force on the beads and (c) magnetic field inside and outside of the magnet, distributed along the magnet symmetry axis as functions of the distance from the tip with magnet core radius of 3, 6, 9, and 12 mm. Magnet tip length is fixed at 10 mm, while the tip radius is 1.5 mm. Magnet core length equals 5 cm.
Figure 9
Figure 9
(a) Magnetic field outside of the magnet, in the vicinity of its tip and (b) magnetic field inside the magnet. Fields are along the magnet symmetry axis. The relative magnetic permittivity of the core material is taking the values of 10, 100, 1000, and 10 000. Magnet tip radius=1.5 mm, tip length=10 mm, core radius=6 mm, core length=5 cm.
Figure 10
Figure 10
Linear response of magnetic B field with current applied to the coil. Hall-effect sensor is located at a distance of 0.5 mm from the tip of the core. Inset: a region of the field-current loop near the origin magnified (10×) to illustrate low remanence observed for the ferromagnetic core.
Figure 11
Figure 11
Comparison of the simulated and experimentally evaluated magnetic B fields in the area of interest below the tip of the electromagnet. Simulated curve represents the vertical component of the magnetic B field, averaged over the circular area of the diameter of 1.2 mm, in order to account for data acquisition by a finite size Hall-effect sensor (1.2×1.2 mm2). The sensing chip is buried inside plastic packaging at the depth specified as 0.5 mm from the surface. Simulation took into the account the actual coil geometry, consisting of seven layers of AWG 30 wires (total of 1359 turns) carrying the current of 0.94 A.
Figure 12
Figure 12
(a) A scheme for the general experimental setup for magnetic tweezers in single molecule stretching experiments. (b) Colorized image of the magnetic bead illuminated by the evanescent wave (532 nm laser) in a TIRF microscope. Upon applying force (F) to the magnetic probe, the intensity of the bead fluorescence changes due to an increase in the distance (z) from the sample surface. (c) Applied electromagnet current and (d) raw probe intensity data (normalized to the intensity observed when no current is applied to the coil) vs time. These data can be converted into force-extension curves with proper calibration.
Figure 13
Figure 13
Individual force-extension curve for 150-mer single stranded DNA obtained from one of the intensity-current cycles in Fig. 12d.
See this image and copyright information in PMC

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