Chapter 6 – Forces 225
a
Estimate the average angle of deviation of laser photons in that optical trap, assuming
that the lateral force arises principally from photons traveling close to the optical axis.
At what frequency for the position fluctuations in the focal plane is the power spectral
density half of its maximum value? (Assume that the viscosity of water at room tem
perature is ~0.001 Pa·s.)
The bead in this trap was coated in titin, bound at the C-terminus of the molecule,
while the bead in the other optical trap was coated by an antibody that would bind to
the molecule’s N-terminus. The two beads were tapped together to try to generate a
single-molecule titin tether between them.
b
If one tether binding event was observed on average once in every ntap tap cycles,
what is the probability of not binding a tethered molecule between the beads?
c
By equating the answer to (b) to Pteth(n = 0) where Pteth(n) is the probability of forming
n tethers between two beads, derive an expression for 〈〉
n in terms of ntap.
We can write the fraction a of “multiple tether” binding events out of all binding
events as Pteth(˃1)/(Pteth(1) + Pteth(˃1)).
d
Use this to derive an expression for α in terms of 〈〉
n .
e
If the bead pair are tapped against each other at a frequency of 1 Hz and the incu
bation conditions have been adjusted to ensure a low molecular surface density for
titin on the beads such that no more than 0.1% of binding events are due to mul
tiple tethers, how long on average would you have to wait before observing the first
tether formed between two tapping beads? (This question is good at illustrating how
tedious some single-molecule experiments can sometimes be!)
Answers
a
Since there is an 80% power loss propagating through the AOD and the laser
beam is then time-shared equally between two optical traps, the power in each
trap is
0 375
0 8 2
0 15
.
. /
.
W
W
(
)×
=
Using Equation 6.3, the angle of deviation can be estimated as
θ =
×
×
×
(
)
×
=
=
−
−
−
(20
N)
ms
W)
40 mrad
2.3o
10
3 10
150
10
12
8
1
3
/ (
Modeling the power spectrum of the bead’s lateral position as a Lorentzian
function indicates that the power will be at its maximum at a frequency of zero,
therefore at half its maximum P
p
v
v
v
v
( )
( ) =
=
+
(
)
/
/
/
.
0
1 2
0
2
2
0
2
Thus, ν = ν0, the
corner frequency, also given by k/2πγ. The optical trap stiffness k is given by
20
10
200
10
1 10
12
9
4
1
×
×
= ×
−
−
−
−
N
m
Nm
The viscous drag γ on a bead of radius r in water of viscosity η is given by 6πrη;
thus, the corner frequency of the optical trap is given by
1 10
2
6
0 89
10
1 10
949
4
6
3
×
×
×
×
× ×
=
−
−
−
π
π
.
Hz
b
The probability of not forming a tether is simply equal to (1 − 1/ntap).
c
Using the Poisson model for tether formation between
<n>nexp(–<n>)/n!