Solutions to Questions and Problems
Frog Antibiotics: Statistical Mechanics of
Helix-Coil Transitions
1. What is an amphipathic protein? The authors refer in their paper to SUVs.
They’re not Hummers, what are they? Sketch a cartoon to amplify your answer.
What are double-D-analogs?
An amphipathic protein is one that has two sides, one that is hydrophobic, one that is
hydrophilic, as shown in the cartoon below
In this context an SUV is a small unilamellar vesicle. As the name indicates, these are
single walled (uni-lamellar) structures made from a 3:1 ratio of two lipids (POPC and
POPG – the longer names are given in the paper on pg 786, under “Results”). Vesicles
formed from lipids that have a polar head and a hydrophobic tail can be schematically
represented as
Double-D analogs are proteins in which two adjacent amino acid residues have been
replaced by the corresponding D-enantiomer.
2. In general, will the formation of a helix be entropically favored? Why or why
not? Why do you think proteins embedded in a membrane are more likely to be
helical? Draw a cartoon of the process (see p. 790) to accompany your answer.
A helix is more ordered than the random coil form, so entropy does not favor its
formation. You could imagine that it would be easier to insert a helical protein into a
membrane than a random coil, as you would have to disrupt less of the membrane to
accommodate the more slender helix.
Helix-Coil Transitions: Solutions to Questions and Problems
The authors comment, however, that the axis of the helix remains parallel to the surface
of the membrane, which suggests that the helix formation aids incorporation of
amphipathic proteins into the membrane by arranging the hydrophobic residues on one
side (the side that is embedded in the membrane) and puts the hydrophilic residues on the
outside.
3. Does Figure 3 in the paper support the contention that the helical form of
magainin 2a is more favorable for binding to the SUVs? Which thermodynamics
parameters favor the helical form in binding, entropy or enthalpy or both?
Yes, the free energy of binding becomes more negative as the %helicity of the protein
increases. The enthalpy term favors binding of helical proteins, while the entropy term
favors the random coil. The enthalpy term dominates.
4. On page 789 [Thermal unfolding and helix-coil theory] the authors study the
helix-coil transition when magainin is bound to the SUVs. What is an isodichroic
point and why does it indicate two states? [Bromberg and Dill is a good place to
look for the answer to this].
An isodichroic (or isobestic) point is where several spectra overlap. For example,
consider the case in which you vary the pH of a solution of some colorimetric indicator
(e.g. the reaction is In + H +  InH + and the two species In and InH+ have different UVvis absorbance spectra). The overlapping spectra would look like
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Solutions to Helix-Coil Transitions
Why does the existence of an isodichroic point suggest that there are two different states
(or species) present? Assume that you have two species; the function describing the
intensity of the absorbance of the first species as the wavelength varies is f(λ) while that
describing the intensity as a function of wavelength for the second species is g(λ). The
overall spectra will depend on the percentage m of each of the two species present
h ( λ ) = mf ( λ ) + (1 − m)g ( λ )
You can see this makes sense, as when the solution is all species 1 (m=100%), the
spectrum h is the same as f, when it’s all species 2 (m=0%), the spectrum h is the same as
g. If the spectra for pure 1 and pure 2 intersect at some wavelength λi, then
f ( λi ) = g ( λi )
What would the value of the spectrum of the mixture be at λi?
h ( λi ) = mf ( λi ) + (1 − m)g ( λi ) , substitute g ( λi ) = f ( λi )
h ( λi ) = mf ( λi ) + (1 − m) f ( λi )
h ( λi ) = mf ( λi ) + f ( λi ) − mf ( λi )
h ( λi ) = f ( λi )
In other words, no matter how much, or little, of species 2 is present in the mixture, the
value of the intensity of the absorbance for the mixture is exactly the same at that one
wavelength, λi. There is an expanded explanation of isodichroic points on page 504 of
Bromberg and Dill.
5. An extremely simple model for the interconversion of coils and helices considers
each residue to be either part of a coil (C) or part of a helix (H) where the
probability of any given residue being a type H (i.e. is part of an α-helix) is
independent of the type of amino acid and independent of whether either neighbor
is a type H or C. Is this likely to be true for most proteins? Why or why not?
Would this model be adequate to describe cooperative helix-coil transitions?
Questions 6-8 use this model to compute an estimate of the entropy of the helix-coil
transition in magainin.
Since the backbone of every amino acid is the same, and the backbone positions are what
make up the framework of the helix or coil, it’s easy to see why this might be a good first
approximation. However, the secondary interactions between residues, which can help
Helix-Coil Transitions: Solutions to Questions and Problems
stabilize one form or another, certainly would depend on the identity of the residue and
on the identity of its neighbors. This type of model would not be able to predict
cooperative transitions, as the presence of an H type residue induces other residues to
conform to a helical shape.
6. Recall that for a single “item” that has two possible states, the partition function
is given by
q = 1+ e−ε / kT
Take the two states to be C and H, let C be the ground state and let H have an
energy of ε. Now write the probability of a single residue being in the H state. Let’s
call the probability of the residue being in the H state ƒH or the fractional helicity.
Let s = e−ε / kT and now write ƒH in terms of s. Find s for a 47-residue polypeptide (like
magainin) that has 83% helicity. Why do we care about s? Go on to question 7 to
find out!
Recall that for any state j the probability of being in the state can be expressed in terms of
the partition function as
− ε kT
e j
pj =
q
The probability of any given residue being in a helical state is
e− ε kT
pH =
1 + e− ε kT
Since s = e−ε / kT we can rewrite pH in terms of s as
s
pH =
1+ s
The fractional helicity is the number in the helical state over the total number of residues,
so we can write
n
p N
s
f H = H = H = pH =
N
N
1+ s
For a 47-residue polypeptide with 83% helicity, s is found by solving
s
fH = 0.83 =
1+ s
s = 4.88
7. Suppose we have an ensemble of polypeptides, each assembled by placing
at total of N residues in a linear sequence, where each residue is assigned to
be a C or H according to the probabilities given by q in question 6. The
ensemble partition function Q is then given by [see Bromberg and Dill p. 500]
Q = (1+ e−ε / kT ) = (1+ s) N
Using the expression for the entropy from statistical mechanics
kT ⎛ ∂Q ⎞
S=
⎜ ⎟ + k lnQ
Q ⎝ ∂T ⎠
N
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Solutions to Helix-Coil Transitions
derive an expression for the absolute entropy S as a function of s for this
model. Compute S for ƒH=0.83 (83% helicity). What are the units on the
entropy you computed?
S=
=
=
kT
(1 + s )
N
kT
(1 + s )
N
1
(1 + s )
N
substituting
1
S=
(1 + s )N
=
∂
1 + e− ε kT
∂T
(
)
N
+ k ln (1 + s )
⎛ ε N − ε kT
1 + e− ε kT
⎜⎝ 2 e
kT
(
⎛ ε N − ε kT
e
1 + e− ε kT
⎜⎝
T
(
)
)
N −1
N −1
N
⎞
N
⎟⎠ + k ln (1 + s )
⎞
N
⎟⎠ + k ln (1 + s )
s = e−ε / kT
⎛ εN
N −1 ⎞
N
s (1 + s ) ⎟ + k ln (1 + s )
⎜⎝
⎠
T
s εN
N
+ k ln (1 + s )
(1 + s ) T
recognize that you can write ε in terms of s ln s = −ε / kT or −kT ln s = ε
s (−kT ln s)N
N
S=
+ k ln (1 + s )
T
(1 + s )
= −Nk
s
ln s + Nk ln (1 + s )
(1 + s )
⎛
⎞
s
= Nk ⎜ ln (1 + s ) −
ln s ⎟
(1 + s ) ⎠
⎝
⎛
⎞
s
The second term in the above expression ⎜ ln (1 + s ) −
ln s ⎟ has no units, so
(1 + s ) ⎠
⎝
the units on S would be those of Nk, in SI J/residue-K.
For a 47 residue polypeptide with 83% helicity, s=4.88 (see question 6), so
⎛
⎞
4.88
S = 47k ⎜ ln (1 + 4.88 ) −
ln 4.88 ⎟ = 21.4k = 2.96 × 10 −22 J/K
(1 + 4.88 )
⎝
⎠
8. The authors say that the magainin forms a “frayed” helix when bound to the
membrane. That is, it’s not all helix even when bound; the ends presumably are still
in random coil. They find that magainin 2 amide (M2a) is 74% helical at 30oC.
Using the equation derived for S in question 7, compute the change in entropy in
going from a partially coiled protein (at 30%) to a more helical form (74% helicity).
Does your answer make sense? Divide by the number of residues to compute the
entropy change (S(“frayed” helix)-S(pure coil)) per residue and compare it to the
value of roughly 2 J/mol-K that the authors measured. How does your model
compare? The actual protein is probably about 10% helical in solution; compute
the entropy change for going from 10% helix to 74% helix. Does this answer make
Helix-Coil Transitions: Solutions to Questions and Problems
sense? You might want to sketch a graph of the entropy as a function of helicity to
see why or why not this model is adequate.
Start by computing s for 10%, 30% and 74% helicity using
s
fH =
1+ s
s(10%)=0.11; s(30%)=0.43; s(74%)=2.85
Now compute absolute entropies for using the expression derived in question 7.
⎛
⎞
s
S = Nk ⎜ ln (1 + s ) −
ln s ⎟
(1 + s ) ⎠
⎝
S(10%)=15.2k; S(30%)=28.7k; S(74%)=26.9k
Now computing the ΔS for the partial coil (30%) to frayed helix (74%) transition gives
ΔS(30%74%)= S(74%)- S(30%)=26.9k-28.7k=-1.8k
The sign of the answer is reasonable; this process should not be favored entropically. On
a per-residue basis this is about 0.3 J/mol-K, different by about an order of magnitude.
If we assumed that the “pure” coil form was about 10% helix, and computed ΔS for the
transition we would find where the theory really frays.
ΔS(10%74%)= S(74%)- S(10%)=26.9k-15.2k=11.7k
Now, even the sign is incorrect!
If we graphed S vs. s we can see that the entropy does not monotonically decrease as the
helical character increases, but instead peaks, then falls off.
9. Why do think the model developed in questions 6-8 fails or succeeds? How does
the Zimm-Bragg model used in the paper to analyze the helix-coil transition of the
bound protein differ from the model developed in questions 6-8 above? Why is the
Zimm-Bragg model likely to be an improvement?
This non-cooperative model assumes that there is single “best” state with some optimal
percent helicity, rather than a coil form and a helix form that are interconverting. The
Zimm-Bragg model is a cooperative, two state model. Since the experimental evidence
indicates the presence of two states, Zimm-Bragg is a reasonable model to use to analyze
the transition.
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