Imaging and Manipulation of Biomolecules - Bionanoelectronics: Bioinquiring and Bioinspired Devices - page 113

where m B is the mass of the cantilever and is the damping coefficient. Considering
that F appl D F exp.i!/, the displacement is z D A exp.i!/, and thus the
frequency response of the cantilever is

A.!/ Dj F j = ! 4 C ! 2 . 2 =m B 2K=m B / C K 2 =m B 1=2 :

(3.12)

Considering that D m B ! 0 =Q, the modulus and the phase of the amplitude are,
respectively,

j A.!/ jD FQ ! 0 =Œ! 2 ! 0 C Q.! 0 ! 2 / 1=2 ;

(3.13)

'.!/ D arctanŒ!! 0 =Q.! 0 ! 2 /:

(3.14)

The first natural resonance frequency is then

! 0 D 2f 0 D .K=m B / 1=2 ;

(3.15)

while the cantilever response is minimum at

! min D .! 0 2 =2m B / 1=2 : (3.16)

Equation ( 3.16 ) tells us that in the damped oscillator model, the frequency response
of the cantilever differs from the natural frequency, the shift between these
frequencies vanishing in the undamped case. Small cantilevers, with dimensions
up to the order of microns, have much higher resonant frequencies (>500 kHz in
air) than larger cantilevers but comparable spring constants (<100 mN m 1 ).
The applications of AFM in biology are huge and are reviewed in Le Grimellec
et al. ( 2010 ). This reference provides also very nice images of various biomolecules.
The AFM was first used for the imaging of soluble molecules such as DNA, with
the aim of investigating their molecular structure ( Lyubchenko et al. 2011 ). DNA-
protein interactions and bindings between DNA polymerase with its DNA target,
as well as the RNA polymerase with its DNA target are studied on bacteria such as
Escherichia coli and on more complex structures. Images of proteins such as GroEL
and GroES of E. coli were seen for the first time. The resolution was enhanced to
1 nm using a cryo-AFM ( Mou et al. 1996 ). Peptides and small proteins, which are
generating fibrils and/or amyloid deposits associated with the Alzheimer disease,
are also studied with AFM ( Adamcik et al. 2010 ). Biological membranes are the
subject of intensive AFM studies ( Frederix et al. 2009 ), as well as cell interactions
with membranes ( Hilal et al. 2006 ). Other applications of AFM in biology are cell
imaging and the study of their mechanical properties and adhesion. In this way,
it was determined that the Young modulus of cells is in the range 0.2-200 kPa
depending on the cell type. These mechanical properties vary if various drugs are
present or if the physiological conditions change. Ligand-receptor pair separation
forces are measured as well with AFM. For example, this force is around 170 pN for
avidin-biotin, 260 pN for streptavidin-biotin, etc. Not only biomolecules, cells and