Chapter 6 – Forces 243
Answers
a
We can first deduce the dimensions of the cantilever: wx = 0.1 mm, wy is equal to
the edge length a of a tetrahedron of height h of 10 μm—tetrahedral trigonom
etry indicates that wy = a = h(3/2)1/2 = 12.2 μm. The volume V of the cantilever is
calculated from mass/density:
V =
×
×
=
×
−
−
(25 10
)/(8.1 10 )
3.1 10
m
12
3
15
3
Thus,
w
V
w w
z
x
y
=
−
=
×
×
×
×
=
−
−
(
)
(3.1 10.15)/(1 10
1.22
10
)
2.5 m
4
5
µ
The resonance frequency equation for a deflecting beam thus indicates
v0
6
4 2
8
3
1/2
0.162
(2.5
10
)/(1 10
)
((2.1 10 )/(8.1 10 ))
=
×
×
×
×
×
×
−
−
= 6521 Hz or ~6.5 kHz
kz =
×
×
×
×
×
×
×
−
−
−
(1/4)
(2.1 10 )
(12.2
10
)
((2.5
10
)/(1 10
))
8
5
6
3
4
= 0.1 Nm
or ~100 pN/nm
1
−
“Stiff” optical tweezers have a typical stiffness of ~0.1 pN/nm, so this AFM canti
lever is ~4 orders of magnitude stiffer.
b
Rearranging the beam equation to give the total effective mass of the cantilever–
tip system indicates
m0
3
0.1/(6.1 10 )2
4
6.8
10 kg
68pg,
11
=
×
×
=
×
=
−
π2
which is three times the
mass of the cantilever. The volume of the tip is given by the tetrahedron formula
of (1/3) × base area × height = 2.1 × 10−16 m3
Thus, the tip mass = (3.4 × 103) × (2.1 × 10−16) = 0.7 pg, which is two orders of
magnitude smaller than the effective mass of the cantilever–tip system and thus
negligible.
c
From Hooke’s law, the change in vertical displacement of the cantilever
corresponding to a 20 nN drop in force is
(20
nm
×
=
−
10
0 1
200
9 ) / .
Assuming sample deformation effects are negligible, this indicates that the
average lateral spacing between pili on the surface is
(8
nm
×
×
×
×
=
−
−
15 10
200
10
155
9
9 1 2
) /
Model the occupancy area around each pilus as a circle of radius 155/2 = 78 nm.
Thus, the mean surface density of pili is
1/(
(78
)
pili per m
π ×
×
=
×
−
10
5 2
10
9 2
13
2
)
.
The range of cell diameters in a population is 1–3 μm, equating to an area range
of 3.1–28.3 μm2. Thus, the range of number of pili in the population is 161–1450
per cell, 1000 PapA molecules per pilus, and therefore the copy number, assuming
all cellular PapA expressed in pili, is 1.6 × 105 to 1.5 × 106 PapA molecules per cell.
d
The tip broadening artifact results in an apparently larger pilus width. The half
angle of tip is ~(180 – 109.5) = 35.3° where 109.5° is the tetrahedral angle, so tip
broadening is tan(35.3°) + sec(35.3°) = 1.9, so the apparent width measured by
AFM imaging of pilus will be ~2.5 × 1.9 = 4.8 nm, as observed.