PES 3950/PHYS 6950: Homework Assignment 4
Handed out:
Due in:
Friday February 20
Friday February 27, at the start of class at 3:05 pm sharp
Show all working and reasoning to receive full points.
Question 1 [5 points]
In class, we derived an expression for the laminar fluid velocity field in a pipe or radius
R and length L, where L >> R. The flow is produced because of a difference in pressure,
P (0) > P (L).
Given the diagram above, the fluid will flow in the z-direction and the velocity field is a
function of r only, ~v = v(r)ẑ. For steady state and in cylindrical coordinates, the NavierStokes equation simplifies to
∂P
1 ∂
∂vz
=η
r
.
∂z
r ∂r
∂r
Solve this differential equation and show that you get
r2 ∆P
+ C1 ln(r) + C2 = −ηvz ,
4L
where ∆P = P1 − P2 . C1 and C2 are constants.
Question 2 [15 points]
a) The lipid bilayer consists of saturated and unsaturated fatty acids. Do you expect unsaturated lipid chain regions to have higher or lower viscosity compared to saturated region?
Explain.
b) Upon hydration, lipids can aggregate and form different structures. For cylindrical aggregates (cylindrical micelles), calculate the expected packing parameter range.
c) You want to make micelles and a lipid bilayer in your lab. You go to the fridge to
check which kind of lipids you have in stock. It turns out that you have lysophosphatidylcholines and phosphatidylcholines. Which type of lipid will produce micelles and which will
give you a bilayer upon hydration? To obtain points you need to explain your reasoning.
[Hint: How many hydrocarbon chains do these lipids have?]
Question 3 [10 points]
Consider the function
h(x1 , x2 ) = x21 + x1 x2 − 2x22 ,
which we assume describes the shape of a deformed lipid bilayer membrane. As shown in
Figure 11.14 (Physical Biology of the Cell), x1 = x and x2 = y are the coordinates of the
reference plane below the membrane.
(a) Compute the two principal radii of curvature. The principal radii of curvature are
the eigenvalues of the matrix of second derivatives, which is
!
∂2h
∂2h
∂x21
∂2h
∂x2 ∂x1
∂x1 ∂x2
∂2h
∂x22
.
√
You should get κ1,2 = ± 10 − 1.
(b) Compute the bending free energy for the piece of membrane corresponding to the square
0 ≤ x1 ≤ 1 and 0 ≤ x2 ≤ 1 in the reference plane. The surface area (SA) is given by
s

2 2
Z Z
∂h
∂h

SA =
+
+ 1 dx1 dx2 .
∂x1
∂x2
At one point you will need to perform a numerical integration to arrive at a bending energy
of 5.2Kb . where Kb is the bending rigidity with typical values in the range 10-20 kB T .
Question 4 [20 points] GRADUATE STUDENTS ONLY
In this equation you are asked to calculate the flagellar cranking torque for a swimming
bacterium.
The Figure above shows a rigid helical rod that is cranked at one end, in such a way that it
sweeps out a cylindrical surface (denoted by the dashed lines). Two short segments of length
l of the rod have been singled out for study, both lying on the near side of the helix and
separated by one turn. The rod is attached (black circle) to a disk and the disk is rotated,
cranking the helix about its axis at angular speed ω. The two short segments then move
downward, in the plane of the page. Thus df~ lies in the plane of the page, but tipped slightly
to the left as shown (see Figure 5.8, Biological Physics). Note that the rod is not free to
translate.
Let P be the pitch, R the radius, and L the total length of the helix. The rod turns through
a full revolution in time T . You may neglect the flow field that the rotating helix sets up in
the surrounding water.
Following the calculation in class, a thin rod segment of length l, pulled parallel (k) to
its axis feels a drag force directed opposite to its velocity. A segment pulled perpendicular
(⊥) to its axis feels a drag force, again directed opposite to its velocity.
a) [10 points] Following the calculation in class with the drag coefficient of γ⊥ ≈ 2γk ≈ 4πηl
for a thin rod segment, where η is the viscosity of water, show that the formula for the total
torque that must be applied to the rod to make it rotate with a given period T is given as
1 2
2
2
sin θ + cos θ ẑ,
~τ = R ωL4πη
2
where θ is the angle that the rod makes with the z-axis. Note, we are cranking the helix
about the z-axis, so we only want the component of torque about this axis. So for each rod
element all we want is the component τz = (~r × f~)z . Moreover, all rod elements make equal
contributions to this τz . Consider one of the unshaded elements in the figure. The vector
~r from the rotation axis to this element points along ŷ, so the only component of the drag
force that we need is fx . Note that the velocity of the selected rod element points along −x̂
as shown in the Figure.
b) Show that the magnitude of the total torque expression can be rewritten as


2 !−1
2πR
.
τz = 2πηR2 ωL 1 + 1 +
P
c) Using values from class, evaluate the total torque τz in water. Compare the torque you
find to the maximum torque that the flagellar motor can exert (the ”stall torque”), which is
around 4000 pN nm. Comment on your finding.