6
1.1 Motivation for Biophysics
DNA molecule if just one molecule of DNA is trapped between the two beads. What
happens if there is more than one DNA molecule tethered between the two beads?
b
It was found that when the center of one of the trapped beads moved from the center
of the focused laser by 10% of its diameter, the measured force on the bead increased
from zero to 3 × 10–11 N, while the distance between the two beads increased from zero
to 1.3 × 10–6 m. How much energy do you think is required to fully extend a molecule
whose fully extended length is 1.5 × 10–5 m? Roughly, how many molecules of water
would be needed to transfer all their kinetic energy to extend the DNA fully? Assume
that the average thermal energy of a molecule of water is of the order of kBT where kB
is the Boltzmann constant (1.38 × 10–23 J/K) and T is the absolute temperature.
c
In practice, when the DNA was continuously stretched, one of the beads was pulled
out of one of the optical traps before the DNA could be fully extended. Any ideas why?
Answers
This question is a little unfair, since it involves some concepts that you have not prop
erly encountered yet in this book—a bit of a spoiler, sorry! But it’s here more to get you
in the mood of thinking about biological systems as physical problems…
a
We’re told that the “extent” of the optical trap is about the same as the wave
length of the laser light. You might hazard a guess and assume that, given no
more information, the optical trap was a spherically symmetrical sphere whose
diameter was the same as the laser’s wavelength. If we assume this, then the max
imum distance from the center to the edge of the trap is half of 1/1,000th of a
millimeter. This 1/1,000th of a millimeter is encountered a lot at the length scale
of cells and is called a micron (symbol µm), equivalent to 1 × 10–6 m. Half of this is
0.5 µm, or 500 nanometers (unit nm—this again you will encounter as the length
scale relevant to single molecules and is equal to one billionth of a meter, 1 ×
10–9m). We’re told that the optical trap itself is a Hookean spring of stiffness k (i.e.,
stress is linearly proportional to strain, or trapping force is proportional to dis
placement of bead from trap center, call it x), so the trapping force is equal to kx.
Maximum force is when x is maximum, so equal to k (0.5 × 10–6) assuming k is in
SI units. With one DNA molecule, this then is the maximum force applied, with n
tethered molecules between the two beads, which we assume are simply acting
in parallel, the maximum force on each is then just k(0.5 × 10–6)/n. Note, in reality,
when a laser is focused, the shape of this diffraction-limited intensity volume of
light in 3D, often referred to as the “confocal volume,” has more of a pigskin or
rugby ball type shape (depending on which side of the Atlantic you are…), which
you will learn about in Chapter 3.
b
The information given here allows you to determine what k is: 10% of the bead
diameter is 300 nm, called this Δx, resulting in a force change, ΔF of 3 × 10–11 N,
or 30 × 10–12 N, or 30 pN, where 1 pN or piconewton is 1 × 10–12 N, and is the typ
ical force scale for single biomolecules in cells. So, k is just dF/dx, or ΔF/Δx for this
Hookean spring case where k is a constant, or 1 × 10–3 N/m—this is also equivalent
to 0.1 pN/nm, where the pN/nm is often a unit of stiffness used in the content of
mechanical experiments on single biomolecules. The energy required to stretch a
Hookean spring a distance L is the work done in moving through a force kx from
x=0 to x=L. This is the integral ∫kx.dx from x=0 to L, or 1/2kL2 since L =1.5 × 10–5 m
(or 15 µm) this is equal to about 1 × 10–13 J. The thermal energy of a water mol
ecule is around kBT (for more details, see Chapter 2), which is roughly the energy
scale of single biomolecules is cells. At room temperature, T is around 300 K, so
kBT is roughly 4.1 × 10–21 J, or 4.1 pN.nm if you like (this can be a useful number
to remember….). This is around 27 million times smaller than the estimate above
for the energy required to stretch a single DNA molecule, and so since kinetic
energy is the same as thermal energy here then 27 million water molecules would