Chapter 6 – Forces 261
indicated protein topographic features 0.5 nm pointing above the bilayer. When the
AFM tip was pushed into these features and then retracted, it was found that the tip
experienced an attractive force toward the membrane. When the same experiment
was performed using a living cell, similar topographic features could be imaged, but
when the tip was pushed into the sample with the same force limit set as before and
then retracted, no such pulling force was experienced. Explain these observations.
6.14 An AFM image was obtained for a hard spherical nanoparticle surface marker
between live cells stuck to a mica surface. The image obtained for the nanoparticle
did not indicate a sphere but a hump shape whose width was ~150 nm larger than its
estimates obtained from transmission electron microscopy.
a
Explain this.
The AFM tip was a tetrahedron with a base edge length of 900 nm and a base tip
height of 10,000 nm.
b
What is the diameter of the nanoparticle?
6.15 Physiological “Ringer” solution has a resistivity of 80 Ω·cm. What is the total electrical
resistance measured across a typical open sodium ion channel of length 5 nm and
pore diameter 0.6 nm?
6.16 A silicon-substrate nanopore of 5 nm diameter was used to detect the translocation of
a polymeric protein in the pH buffer “PBS” using a 120 mV voltage across the nanopore.
The protein consists of 5 α-helices containing 10–20 amino acids each connected by
a random coil of 5–10 amino acids. The protein had a small net positive charge and
it was found that there were just two cysteine residues separated by 20 amino acids.
When the electric current through the nanopore was measured, it indicated that for
most of the time the current had reasonably stable value of 50 pA, but also had much
shorter-lived 40, 42, 44, and 46 pA. However, when 5 mM DTT (see Chapter 2) was
added to the solution the short-lived current values were measured at 40, 42, 44, 46,
and 48 pA. Explain these results.
6.17 Graphene is a very thin yet strong structure and also electrically conducting. Is this an
advantage or disadvantage to using it as the nanopore substrate for sequencing single
DNA molecules?
6.18 Fick’s first law of diffusion (see Chapter 8) states that the vector particle flux J = −D ·
grad(n) where D is the diffusion coefficient and n is the number of particles per unit
volume.
a
Modeling an ion channel as a 1D cylinder of radius a, derive an expression for the
channel current due solely to diffusion of univalent ions of molar concentration
C, stating any assumptions you make.
b
In a patch clamp experiment, an extracted region of cell membrane contained
~10 Na+ ion channels each of diameter 1 nm. When a voltage of −150 mV was
applied across the membrane patch in a solution of 175 mM NaCl, the measured
current was found to fluctuate with time from a range of zero up to a maximum at
which the observed resistance of the patch was measured as 2.5 × 109 Ω.
c
Estimate the current through a single Na+ channel and the minimum sampling
frequency required to monitor the passage of a single ion. How significant is
diffusion to ion flux through a single channel?
6.19 A cell was placed in a physiological solution consisting of 100 mM NaCl, 20 mM KCl
at room temperature. The cell membrane had several open Cl− channels; using single-
molecule fluorescence imaging, their internal concentration of Cl− ions was measured
at 20 mM, while that of K+ was 30 mM.
a
What is the transmembrane voltage on the basis of the Cl− concentration? Why is
it sensible to use Cl− concentrations for this calculation and not K+?
b
It was found that K+ would on average not spontaneously translocate out of the
cell, but rather that this required energy to pump K+ out. Why is this?
c
A chemical decoupler was applied that forced all Na+ and K+ ion channels to open,
and the ions then moved across the membrane to reach electrochemical equilib
rium. Would you expect the K+ ion concentration inside and outside the cell to
be equal?