Lab Manual: Limits of Detection

Overview

Atomic force microscopy and optical trapping are key techniques for investigating forces in biological systems at cellular and molecular levels. Both instruments can exert and measure forces on submicron-scale particles. This capability offers a unique and valuable tool for manipulating and measuring cell components at the single molecule level. For example, optical traps have been used extensively to investigate the mechanical properties of biological polymers and the force generation mechanisms of molecular motors. In many studies, optical tweezers apply force to functionalized microspheres, which act as convenient handles attached to molecules of interest.

To make quantitative force measurements, AFMs and optical traps record displacement of objects being measured versus time. For small displacements, the exerted force is very nearly proportional to displacement. The force/displacement relationship is modeled by Hooke's law: F = -αx, where α is the spring constant. The detector produces a voltage proportional to displacement. The constant of proportionality that converts voltage to distance is called the responsivity of the instrument and denoted by the letter R. Both α and R must be found through a process called calibration. Accurate force and position measurements depend on careful calibration of R and α. On the AFM, α is a function of the cantilever geometry. The optical trap's stiffness depends on trapping laser power, bead size, bead composition, and optical properties of the sample.

In this lab, you will investigate the limits of force detection using AFM and optical traps. You must attend a one-hour lab session during which an instructor will help you take data. You will analyze the data to determine α and R, and you will calculate the minimum detectable force, δ, for both instruments.

Background reading

• Neuman & Nagy. Single-molecule force spectroscopy: optical tweezers, magnetic tweezers and atomic force microscopy. Nature Methods - 5, 491 - 505 (2008).

Optical trap procedure

The optical trap software produces four columns of tab-delimited data. The first two columns are the X- and Y-axis sample stage readouts in Volts. To convert the voltage to position, use the responsivity of the stage, R_stage = 1/2.2  μm/V. The third and fourth columns are the X- and Y-QPD voltages. Use the QPD responsivity, R, to convert the QPD voltages to position.

The calibration methods are summarized below:

Trap stiffness calibration methods and sensitivity to measurement parameters

Method	Equation	QPD Responsivity	Stage Responsivity	Solvent Viscosity	Particle Diameter	Temperature	Technical Noise
		$ R_{QPD} $	$ R_{stage} $	$ \eta $	$ d $	$ T $
Equipartition	$ \frac{K_B T}{\langle R_{qpd} V_{qpd} \rangle ^ 2} $	inverse square	none	none	none	linear and indirect (viscosity change)	systematic decrease
PSD	$ \left. {6 \pi^2 \eta d \, f_0} \right. $	none	none	linear	linear	indirect (viscosity change)	small
Stokes	$ \langle \frac{3 \pi \eta d \, R_{stage} \, ^{d V_{stage}} / _{dt}} {R_{qpd} V_{qpd}} \rangle $	inverse	linear	linear	linear	indirect (viscosity change)	none

Report requirements

• Optical trap
  • Find R for each power setting by the PSD method.
  • Find α for each laser power setting by the Stokes, equipartition, and PSD method.
  • Plot R versus laser power for the PSD method.
  • Plot α versus laser power for all three methods.
  • Plot δ versus laser power.
  • Calculate $ \delta_x $ and $ \delta_f $ for each laser power setting.
• AFM
  • Calculate the PSD from the voltage vs time data
  • Find the photodiode responsivity R (or lever sensitivity S) from the calibration data
  • Find the lever stiffness k either from the PSD you calculated above or from the averaged PSD
    • Note that to use either method, you have to learn how a PSD is calculated in Matlab (talk to an instructor if needed)
    • How far would the cantilever deflect if you placed a paperclip on the end? (Hint: If you don't want to guess or google the weight of a paperclip, we have a scale in the lab.)
  • Calculate $ \delta_z $ and $ \delta_f $ for the AFM cantilever in air