Determining Trap Compliances, Microsphere Size Variations, and Response Linearities in Single DNA Molecule Elasticity Measurements with Optical Tweezers

Abstract

We previously introduced the use of DNA molecules for calibration of biophysical force and displacement measurements with optical tweezers. Force and length scale factors can be determined from measurements of DNA stretching. Trap compliance can be determined by fitting the data to a nonlinear DNA elasticity model, however, noise/drift/offsets in the measurement can affect the reliability of this determination. Here we demonstrate a more robust method that uses a linear approximation for DNA elasticity applied to high force range (25-45 pN) data. We show that this method can be used to assess how small variations in microsphere sizes affect DNA length measurements and demonstrate methods for correcting for these errors. We further show that these measurements can be used to check assumed linearities of system responses. Finally, we demonstrate methods combining microsphere imaging and DNA stretching to check the compliance and positioning of individual traps.

Keywords: DNA elasticity; calibration; force; laser tweezers; microsphere size; optical trap; single-molecule; trap stiffness.

FIGURE 1

(A) Plots of the magnitudes of the two force-dependent terms in Eq. (1); the linear term in blue. The square-root term (black) can be accurately approximated by a line over the range from F = 25–45 pN (red dashed line). (B) % error made in the square-root term by the linear approximation. (C) Schematic illustration of the variables involved in force-extension measurements of DNA stretched between two optically trapped microspheres. The distance between the trap centers is d, the end-to-end extension of the DNA is x, the radii of the microspheres are r ₁ and r ₂, the force exerted by the tensioned DNA on the microspheres is F, and the displacements of the microspheres from the trap centers are Δx ₁ and Δx ₂.

FIGURE 2

(A) Examples of plots of separation between the traps d = β(V _mirror−V _overlap) vs. F for data recorded when stretching DNA molecules between the two optically trapped microspheres. The points are experimental measurements and the lines are fits to Eq. (4), used to determine trap compliances, microsphere size variations, and average displacement offset factor. (B) Histogram of variations in ϵ values determined by the linear fits of the DNA stretching data to Eq. (4), which characterizes the effect of variations in microsphere sizes.

FIGURE 3

Analyses of the predicted effects of nonlinear errors in system responses. (A) The blue points are recorded force vs. separation data and the blue line shows a linear fit, which describes the data well. The red points predict the effect of a quadratic error term affecting the separation control, which causes detectible curvature in the plot. The red line is a linear fit to these points. Similarly, the green points predict the effect of a quadratic error term affecting the force determination, which again causes detectible curvature. The green line is a linear fit to these points. (B) The blue points are recorded force vs. DNA extension measurements and the blue line shows a linear fit. The red points predict the effect of a quadratic error in the displacement of the microspheres from the trap centers (Δx), which causes detectible curvature, and the red line is a linear fit to these points.

FIGURE 4

(A) Microsphere position, determined by image centroid tracking, vs. mirror control signal (blue points). The red line shows a linear fit. (B) Example of a measured DNA detachment event where the separation of the traps is increased (by increasing V _mirror) and the force is measured to suddenly drop to zero. (C) Measurements of the movement of the microsphere back to the trap center, determined by image centroid tracking, after DNA detachment events that occurred at different force levels. The red line is a linear fit to the data.