242 6.5  Scanning Probe Microscopy and Force Spectroscopy

different frequencies close to the resonance frequency of the tip–​cantilever system. These

waves are propagated into the sample and then result in an acoustic interference pattern. The

acoustic vibrations from the interference pattern are picked up by the tip and transmitted

into the cantilever, either through the sample media of air or water or through direct contact

with the sample in the case of vacuum imaging. Cantilever oscillations are then detected in

the same manner as for AFM using laser beam reflected and imaged onto a split photodiode.

Greatest sensitivity is achieved using force acoustic frequencies slightly higher than the

resonance frequency of the normal flexure mode of the tip–​cantilever system, usually from

~10 kHz up to ~5 MHz. The perturbations to both phase and amplitude across the sample

surface in the acoustic standing wave are locally monitored by the tip, which acts as an

acoustic antenna via a lock-​in amplifier. Note that there are many common standard forms of

instrumentation used in biophysics, which we will not explore in depth of this book; however,

the lock-​in amplifier is of particular use and is singled out here. It is an electronic amplifier

common to many other applications of biophysics that can pull out and amplify a signal from

a very specific frequency carrier wave from an otherwise extremely noisy environment, often

in cases where the signal amplitude is up to six orders of magnitude smaller than the typical

noise amplitude. As such it has myriad uses in single-​molecule biophysics signal detection

and amplification in particular.

Monitoring perturbations of the AFM tip in this way not only generates topographical

information from the sample but also is a direct measure of elastic properties. For example, it

can be used to infer the Young’s modulus deeper in the sample in the region directly below the

scanned tip, with an effective spatial resolution of 10–​100 nm (Shekhawat and Dravid, 2005).

Biological applications have included the imaging of malarial parasites buried deep inside

red blood cells and monitoring aggregation effects in vitro of amyloid peptides (important

precursors of various forms of dementia when misfolded, see Chapter 2).

Worked Case Example 6.2: AFM Imaging

In an AFM imaging experiment, the cantilever–​tip system was composed of a tetrahedral-​

shaped silicon nitride tip of height 10 μm with tip end radius of curvature 15 nm, which

was fixed to a cantilever of mass 25 pg, such that the tip was located at the end of the

cantilever’s 0.1 mm long axis. The cantilever width was equal to the tetrahedron tip edge

length.

a

What is the value of the cantilever resonance frequency in kHz and its stiffness in units

of pN/​nm? Comment on how these compare with the stiffness of typical “stiff” optical

tweezers.

b

How does the effective mass of the AFM tip–​cantilever system compare with the mass

of the cantilever and the mass of the tip?

  AFM imaging was performed using this tip–​cantilever system in contact mode to

investigate an array of spike structures called “pili” (singular =​ “pilus”) covering the

surface of a spherical bacterial cell of ~2 μm diameter. Prior SEM imaging suggested

that pili have a mean ~1 μm length and are composed of ~1000 copies of a sub­

unit protein called “PapA,” with pili expressed over the surface uniformly with a sur­

face density that did not depend on the cell size. When the tip was scanned between

consecutive pilus spikes, the estimated vertical force on the cantilever dropped by

~20 nN.

c

If the range of cell diameters in a population is ~1–​3 μm, estimate the range in PapA

copy numbers per cell, stating assumptions you make.

d

The estimated mean width of a single pilus from earlier SEM imaging was 2.5 nm;

however, AFM imaging suggested a width of almost 5 nm. Explain this discrepancy.

(You may assume that the density and Young’s modules of steel are 8.1 × 10³ kg m⁻³ and

210 GPa, and the density of silicon nitride is 3.4 × 10³ kg m⁻³.)

KEY BIOLOGICAL

APPLICATIONS: AFM

Imaging biological surface top­

ography; Measuring molecular

viscoelasticity and mechanics.