Fig. 4

Linear force-extension relationship enables measuring external forces. a Magnetic bead dynamics undergoing a transition from diffusive (Additional file 4: Fig. S4) to directed motion with increasing, current-controlled external field. b Magnetic force calibration via frictional dissipation. Data averaged over 5 repeats, each lasting ca. 45 s. The errors in the measured forces are contained within the marker size and are negligible compared to the variation in the current used to operate the electromagnetic needle. c Steady-state position of the trapped bead in the absence (left) and presence (right) of an external magnetic force. d Bead displacement from the stagnation point upon the application of varying magnetic fields. The exponential approach could be readily fitted onto a master curve (Additional file 4: Fig. S5). e Linear force–displacement relationship with a spring constant of k = 35 ± 4 fN/\(\mu\)m. Shaded regions and error bars in all plots represent standard deviations