Alpha helices are often embedded in a protein so that one side faces the surface and the other
side faces the interior. Such helices are often termed amphipathic because the surface side is
hydrophilic and the interior side is hydrophobic. A simple way to decide whether a sequence
of aminoacids might form an amphipathic helix is to arrange the aminoacids around what is
known as “helix-wheel projection”, which is a 2d projection of the aminoacid positions (see
Fig. 1). If the hydrophobic and hydrophilic aminoacids are segregated on opposite sides of
the wheel, the helix is amphipathic. Using the helix-wheel projection, decide which of the
three peptides here might form an amphipathic helix
1. S L I K S V I E M V D E W F R T F L
2. F L I R V L R K V F R V L T R I L S
3. R L F R S R V L K I A V I R F L L I
(underlined aminoacids are hydrophobic one, they can be memorized by the mnemonic
“FAMILY VW”).
1.2 Amphipatic beta sheets
Like alpha helices, beta sheets often have a one side facing the surface of the protein and one
side facing the interior, giving rise to an amphipathic sheet with one hydrophobic surface
and one hydrophilic surface. From the sequences listed below pick the one that could form
a strand in an amphipathic beta sheet.
1. A L S C D V E T Y W L I
2. D K L V T S I A R E F M
3. D S E T K N A V F L I L
4. T L N I S F Q M E L D V
5. V L E F M D I A S V L D
1.3 Buried Aminoacids
Small proteins may have only one or two amino acid side chains that are totally inaccessible
to solvent. Even in large proteins, only about 15% of the amino acids are fully buried. A list
Figure 1: You can use this figure to decide which of the three sequences might form an
amphipatic helix (problem 1.1).
of buried side chains from a study of twelve proteins is shown in the table below. The list is
ordered by the proportion of amino acids of each type that are fully buried. What types of
amino acids are most commonly buried? Least commonly buried? Are there any surprises
on this list?
Aa prop. buried Aa prop. buried
I 0.60 S 0.22
V 0.54 E 0.18
C 0.50 P 0.18
F 0.50 H 0.17
L 0.45 D 0.15
M 0.40 Y 0.15
A 0.38 N 0.12
G 0.36 Q 0.07
W 0.27 K 0.03
T 0.23 R 0.01
2 Ideal Flexible Polymers
2.1 Radius of gyration
For an ideal polymer of length N we have shown that the average end-to-end distance R~ is:
hR~ 2
i = a2
N (1)
where a is the average bond length. We define now the radius of gyration as follows:
Rg =
vuut
1
N + 1
X
N
i=0
(~ri ~rcm)
2 (2)
where ~ri is the position of the monomer i (i = 0, 1,…N) and where the center of mass
position is given by:
~rcm = 1
N + 1
X
N
i=0
~ri (3)
a) Show that
R2
g = 1
2(N + 1)2
X
N
i=0
X
N
j=0
(~ri ~rj )
2 (4)
b) Use the previous formula to calculate the average1 hR2
gi
c) Show that in the limit N ! 1 one has
hR2
gi ⇠ a2N
6 (5)
so that the asymptotic N dependence behavior is similar to that of the end-to-end
distance of Eq. (72).
1You will need to use the following relations
X
N
k=1
k = N(N + 1)
2
X
N
k=1
k2 = N(N + 1)(2N + 1)
6
2.2 The Gaussian Chain
We have shown that in general the end-to-end distance for an ideal polymer at large N is a
gaussian distribution
P(R, N ~ ) = ✓ 3
2⇡N a2
◆3/2
exp
3R~ 2
2N a2
!
(6)
irrespectively of the type of model considered. In the calculation we have expanded the
Fourier transform of g(~u) for small k. This approximation is not necessary for a Gaussian
chain, which is defined by the following bond distribution:
g(~u) = ✓ 3
2⇡a2
◆3/2
e
3~u2
2a2 (7)
Show that for a Gaussian chain the probability distribution function can be computed exactly
for any values of N, not necessarily large, and show that the result of the calculation gives
Eq. (6).
2.3 One dimensional polymer
We consider the following one dimensional model of a lattice polymer
g(u) = 1
2 ((u + a) + (u a))
Let us suppose the first monomer is placed in x = 0. Since this is a lattice polymer, the
monomers can only occupy discrete positions x = na with n an integer. The end point will
be in R = ma. It is then more convenient to write the end distribution as P(m, N).
a) Calculate the distribution P(m, N) for every value of N.
b) Show that in the limit N m and N and m both large P(m, N) reduces to a gaussian
which agrees with the general expression derived in the course.
