STRUCTURE O FUNCTION O BIOINFORMATICS
The helix–coil transition revisited
Yantao Chen,1Yaoqi Zhou,2,3and Jiandong Ding1*
1Department of Macromolecular Science, Key Laboratory of Molecular Engineering of Polymers of Ministry of Education,
Advanced Materials Laboratory, Fudan University, Shanghai 200433, China
2Department of Physiology and Biophysics, Howard Hughes Medical Institute Center for Single Molecule Biophysics,
State University of New York at Buffalo, Buffalo, New York 14214
3Indiana University School of Informatics and Center for Computational Biology and Bioinformatics,
Indiana University School of Medicine, Walker Plaza, Indianapolis, Indiana 46202
a-helix, first proposed by Pauling and coworkers in 1950,1,2is one of
the basic structural elements in proteins.3So far, it has been used as a
model system for studying the formation of secondary structures in pro-
teins by experimental,4–13theoretical,14–25and simulation meth-
ods.26–51Theories of helix–coil transitions have been used for interpret-
ing experimental and simulation results. One of the best-known theories
for the helix–coil transition was established by Zimm and Bragg15since
five decades ago. In this classical theory, each residue is assumed to be
either in a coiled or a helical state. The formation of a native hydrogen
bond (H-bond) distinguishes a ‘‘helical’’ residue from a ‘‘coil’’ residue, as
schematically shown in Figure 1. The theory has two parameters. A
nucleation parameter (r) reflects the degree of difficulty in nucleating a
helical residue among coil blocks and a propagation parameter (s) meas-
ures the ability of helix propagation at the helix–coil interface.
These two Zimm–Bragg parameters (r and s) have been considered as
the fundamental parameters for describing the helix–coil transition and
the tendency of different residues in helix formation.5–9For example,
Scheraga and coworkers first put forward a host-guest method, in which
the amino acid of interest is incorporated into long random copolymers
to measure Zimm–Bragg parameters.5Later, an alternative method was
proposed by Baldwin and coworkers, in which the parameters are deter-
mined via substitution into short polyalanine helices.9These experimen-
tal results revealed different tendencies of helical formations for different
residues. The two parameters have also been determined by computer
simulations.27–32,38,40,46,47,49,51Recently, the importance of these pa-
rameters was reassessed byBaldwin.52
Grant sponsor: NSF of China; Grant numbers: 50533010, 20574013, Two-Base Grant; Grant sponsor: Key
Grant of Chinese Ministry of Education; Grant number: 305004; Grant sponsor: 973 Project; Grant number:
2005CB522700; Grant sponsor: Science and Technology Developing Foundation of Shanghai; Grant number:
055207082; Grant sponsor: 863 Project from Chinese Ministry of Science and Technology; Grant sponsor:
Innovation Foundation for Graduate Students from Fudan University; Grant number: CQH1717007; Grant
sponsor: NIH; Grant numbers: R01 GM 966049; R01 GM 068530
*Correspondence to: Jiandong Ding, Department of Macromolecular Science, Key Laboratory of Molecular
Engineering of Polymers of Ministry of Education, Advanced Materials Laboratory, Fudan University,
Shanghai 200433, China. E-mail: firstname.lastname@example.org
Received 23 November 2006; Revised 9 February 2007; Accepted 5 March 2007
Published online 27 June 2007 in Wiley InterScience (www.interscience.wiley.com).
In this article, we perform a dynamic Monte
Carlo simulation study of the helix–coil tran-
sition by using a bond-fluctuation lattice
model. The results of the simulations are com-
pared with those predicted by the Zimm–
Bragg statistical thermodynamic theory with
propagation and nucleation parameters deter-
mined from simulation data. The Zimm–
Bragg theory provides a satisfactory descrip-
tion of the helix–coil transition of a homopo-
lypeptide chain of 32 residues (N 5 32). For
such a medium-length chain, however, the an-
alytical equation based on a widely-used
large-N approximation to the Zimm–Bragg
theory is not suitable to predict the average
length of helical blocks at low temperatures
when helicity is high. We propose an analyti-
cal large-eigenvalue (k) approximation. The
new equation yields a significantly improved
agreement on the average helix-block length
with the original Zimm–Bragg theory for both
medium and long chain lengths in the entire
temperature range. Nevertheless, even the
original Zimm–Bragg theory does not provide
an accurate description of helix–coil transi-
tion for longer chains. We assume that the sin-
gle-residue nucleation of helix formation as
suggested in the original Zimm–Bragg model
might be responsible for this deviation. A
mechanism of nucleation by a short helical
block is proposed by us and provides a signifi-
cantly improved agreement with our simula-
Proteins 2007; 69:58–68.
C 2007 Wiley-Liss, Inc.
Key words: Zimm–Bragg theory; helix–coil
transition; dynamic Monte Carlo simula-
tion; lattice model; nucleation; protein fold-
C 2007 WILEY-LISS, INC.
In this work, a dynamic Monte Carlo (MC) simulation
has been performed to investigate the helix–coil transi-
tion of a self-avoiding lattice chain in the three-dimen-
sional space. Statistical analysis of equilibrium constants
for the dynamic processes of helical block formation and
disruption allows us to directly calculate r and s parame-
ters in the simulation27,51in addition to evaluation of
helicity y and average helical block length L. The avail-
ability of r, s, y, and L from our simulations permits a
detailed comparison between the simulated data and the
predictions given by the Zimm–Bragg theory. We show
that it is necessary to introduce a short helical block
(rather than a single helical residue) for nucleation in
order to bridge the gap between theoretical predictions
and simulation results for chains longer than 32 residues.
Moreover, we find a large-eigenvalue (k) approximation
that leads to a more accurate simple analytic equation of
the average length of helix blocks.
Schematic representation of the formation of an a-helix block. Symbols ‘‘0’’ and ‘‘1’’ denote the coiled residue and helical residue, respectively. K1, K2, and Kndenote the
equilibrium constants of the conversion between the coiled state and the helical state of the interface residue between a helical block and a coil block. The associated
representative snapshots from our simulations of a homopolypeptide are shown on the right side. The helical and coiled residues are colored as red and gray, respectively.
A blue arrow denotes a native H-bonding between the two residues that are four residues from each other in the sequence. A green arrow denotes a nonnative H-bonding.
Here, residue i is denoted as state ‘‘1’’ (a helical residue) only when a native H-bonding with residue i ? 4 is formed.
The Helix–Coil Transition Revisited
The Zimm—Bragg theory
The partition function Q in the Zimm–Bragg theory15
is expressed as53
where Xj,kis the total number of ways to have k helical
hydrogen bonds in j distinct a-helical blocks. Note that a
helical block is made of consecutive unbroken helical res-
idues and N ? 4 is the maximum number of a-helical
hydrogen bonds in a polypeptide chain of length N. In
this theory, the nucleation process is the formation of
the first a-helical residue (the first native hydrogen
bond) among coil blocks and its equilibrium constant K1
is the product of the nucleation parameter and the prop-
agation parameter. The propagation process, on the other
hand, is the formation of the helical residues that are the
neighbors of existing helical residues. The equilibrium
constants for adding and removing a helical residue from
an existing helical block of length n (Kn, n > 1) are
assumed to be the same regardless the length of existing
helical block. So,
K2¼ K3¼ ??? ¼ Kn¼ ??? ¼ s:
For this simple model, Q has an analytical form given by
ðk0? sÞ ? kN?3
where k0;1¼ s þ 1 ?
and k1 are the large and the small eigenvalues of the
characteristic Zimm–Bragg matrix (2l3 2l), respectively.
Here, l ¼ 1 for local interactions between two adjacent
Once Q is known, the helicity or the ratio of the num-
ber of helical residues in N ? 4 residues, y, the average
number of a-helical blocks, m, (helical segments sepa-
rated by coiled residues), and the number average length
of a-helical blocks, L, can be obtained from
ð1 ? sÞ2þ 4rs
=2, k0> k1, k0,
N ? 4
m ¼@ lnQ
L ¼ ðN ? 4Þu
Here, Eqs. (5) and (6) correspond to Eqs. (2) and (44),
respectively, in the original paper of Zimm and Bragg.15
Using Eq. (4), one can obtain
The equations above are bit complicated. The large-N
approximation15(i.e., ln Q ? N lnk0) was proposed to
simplify the equations and leads to15,30,53
1 ? s
ð1 ? sÞ2þ 4rs
L ¼ 1 þ
1 ? s þð1 ? sÞ2þ 4rs
It is of interest to check if the above equations satisfy
the high-helicity limit at low temperatures. At this limit,
it is very easy to propagate once the nucleation process is
completed. That is, s ? 1 and, of course, also s ? r
since r < 1 in all cases. According to Eqs. (11) and (12),
y ? 1 and L ? 1 þ (s ? 1)/r at this limit. While y
approaches the correct limit, the correct limit for L
should be N ? 4. Thus, the large-N approximation is
just appropriate for y but not for L because 1 þ (s ? 1)/
r can be significantly greater than N at low tempera-
tures. In the next subsection, we shall make another
approximation for simplifying those equations.
The large-k approximation
To achieve the correct limit for L at s ? 1 and ?r.
We introduce a large-eigenvalue (k) approximation. A
naı ¨ve way is to neglect all of terms about k1 in the
Y. Chen et al.
expression of Q. Then, Eq. (4) becomes
lnQ ? ðN ? 4Þ lnk0þ lnðk0? sÞ:
This leads to [cf. Eq. (6)].
N ? 4
þ s þ 1þ
? s þ 1;
D ? ð1 ? sÞ2þ 4rs:
Combining Eqs. (11) and (14), we obtain
L ¼N ? 4
N ? 3
2s ? 2s=ðN ? 3Þ
þ 1 ? s þ 2s=ðN ? 3Þ
ð1 ? sÞ2þ 4rs
In this equation, L ? N ? 4 when s ? 1 and ?Nr.
The equation can be further reduced to Eq. (12) if N ?
3 þ (s ? 1)/r or simply if N ? s/r.
Equations (11) and (16) could also be obtained by a
direct simplification of Eqs. (8) and (10) under the
dure. In fact, the so-called large-N approximation in the
literature and the so-called large-k approximation in this
article both contain k0? k1and N ? 1. But the large-
N approximation is stronger than the large-k approxima-
tion. At low temperatures, s ? 1 and r ? 1. Then, N ?
s/r is only justified for an extremely long chain. As a
result, the large-k approximation is useful for both a me-
dium chain and a long chain, whereas the large-N
approximation is just for a very long chain at the high
N?3with a bit tedious proce-
Nucleation by short helical block
In the Zimm–Bragg theory, nucleation is completed af-
ter the formation of the first helical residue and the rate
of propagation is the same regardless the length of the ini-
tial helical block. We think that it is possible that nuclea-
tion requires more than one helical residue. In that case,
Knwill depend on the length of the existing helical block
(n) at least for the first few helical residues (short helical
block). To take into account this possibility, we define
Knþ1¼ Knþ2¼ ??? ¼ s;
where n is the size of the nucleation helix block. The
nucleation parameter rblockfor helical block is given by
Here, we extend the Zimm–Bragg theory by using the
block nucleation parameter rblockand the propagation
parameter s obtained from the equilibrium constant Ki,
i > n. Equations from the original Zimm–Bragg theory,
the large-N approximation, and large-k approximation
can be used to calculate helical ratio and the average
length of helical blocks.
employed to investigate the helix–coil transition. We use
a suitable lattice model because it allows efficient calcula-
tions while retaining essential physics of the transi-
tion.36, 54–58This model is an improved version of a
lattice model developed previously by us.51
We consider a single polypeptide chain of length (N)
of 32 amino-acid residues, unless otherwise indicated.
The lattice space is composed of 36 3 36 3 36 cubic lat-
tice sites and the periodic boundary condition is
set along each dimension. In our model, each residue is
represented by a cube made of eight lattice sites sur-
rounding its center (Fig. 2). The bond length between
two consecutive residues may fluctuate within a defined
range.59There are 87 bond orientations (relative orienta-
tions between cubes) in three dimensions that permit a
large number of possible bond angles.59This bond-fluc-
tuation lattice model59–61allows a better account for the
excluded volume effect and a more accurate description
of helical geometry without significantly changing the ef-
ficiency associated with simple lattice models.62,63An al-
ternative approach to improve the accuracy of lattice
models was the 210 lattice model developed by Kolinski
and Skolnick.64–66A large number of bond orientations
reduce the artificial anisotropy of a lattice model.
The total residue–residue interaction energy E is the
sum of the internal energies for bond length (El), bond
angle (Eu), chirality (Ech) and nonbonded interaction for
hydrogen bond (H-bond, EHB).
work, a coarse-grained lattice model is
E ¼ Elþ Euþ Echþ EHB;
The Helix–Coil Transition Revisited
Here kl and kuare the corresponding force constants
for the associated harmonic potentials; liand yiare the
ith bond length and bond angle, respectively; y0and l0
are their respective expected values. y0 is set to be
88.98 as the actual value in a-helix. All lengths are
scaled by the size of cube. The bond length l0is set to
be6 , the distance between (0,0,0) and (2,1,1). This dis-
tance allows the bonded chain to move efficiently. In this
coarse-grained model, the side chains of amino-acid resi-
dues are not treated explicitly, thus the specific packing
effect cannot be reproduced. As an alternative, we
include a term Echto reproduce the fact that most heli-
ces in proteins are right-handed helices.
ðli3 liþ1? liþ2? 0Þ
ðli3 liþ1? liþ2> 0Þ
That is, a left-handed chirality for three continuous
bond vectors will be penalized with a positive energy ech.
The interaction energy of H-bonding is expressed as
ðh1? hi;j? h2Þ
j ? i þ 4:
Here, eHBis the energy of each H-bond, and hi,jis the
distance between residue i to residue j (The position of a
residue is defined by the residue center). The H-bonding
interaction occurs when the inter-residue distance is
within the distance range from h1 to h2 and their
sequence separation is four residues or more. This model
allows nonnative H-bonds for sequence separation of five
residues or more. The model is an improvement over the
previous model which limits the H-bonds for |i?j| ¼ 4
(i.e. native H-bonds only).51Values for h1and h2as well
as ech, eHB, kl, and kuare determined by trials and errors.
The parameters are set so that helix conformation is sta-
ble at low temperatures. h1¼
eHB/2, kl¼ eHB, and ku¼ 2eHB.
The bond-fluctuation model is simulated with the
same dynamic MC simulation technique employed in
our previous work.51Every MC step contains N
attempts. In each attempt, a residue and four sites that
are the nearest neighbors to one of its six cube surfaces
are selected randomly (Fig. 2).51,59,60The residue
moves when the selected four sites become the new cube
surface of the residue. This move is accepted if all the
four sites are not occupied and the standard Metropolis
Equilibrium constants of helix–coil
transitions from simulations
We use ‘‘0’’ and ‘‘1’’ to denote the coiled and helical
residues, respectively. As shown in Figure 1, the i-th resi-
due is labeled as ‘‘1’’ if an a-helical H-bond is formed
between the i-th residue and the (i ? 4)-th residue and
‘‘0’’ otherwise. (To avoid redundancy, we use only one
direction to define helical
‘‘100 ? 110’’ to denote the propagation of a helix regard-
less the length of a-helical block by converting a coiled
residue located between a helical residue and a coiled res-
idue to a helical one. n100?110,tryand n100?110,sucare the
number of tried and the number of successful attempts
residues). We also use
Schematic representation of the coarse-grained lattice model. Each residue
occupies eight lattice sites. Each lattice site can only be occupied by one residue
at one time. The elementary movement is set to allow ‘‘half residue’’ to jump to
the four nearest-neighbor vacant sites along one of the six marked directions. A
hydrogen bond can be formed between two residues with sequence separation of
more than three residues within a cutoff distance.
Y. Chen et al.
for making a conversion. The probability of propagating
p100?110is thus calculated from
here ‘<>’ denotes the average over all residues and asso-
ciated attempts. Similarly, the possibility of successful
attempts for the reverse process (helical to coiled resi-
due), p110?100, can also been calculated. The equilibrium
constant K100$110 between helix propagating and its
reverse process can be calculated from p100?110/p110?100.
Similarly, the equilibrium constant K000$010 between
nucleation and its reverse process and K101$111between
formation and breakage of helical block can be respec-
tively determined by p000?010/p010?000 and p101?111/
p111?101. Here, p000?010, p010?000, p101?111and, p111?101
denote the probabilities for the nucleation process, its
reverse process, formation, and breakage of an a-helical
block, respectively. In the Zimm–Bragg theory, K100$110
¼ s, K000$010¼ rs, K101$111¼ s/r. Thus, the knowl-
edge of any two equilibrium constants will allow us to
calculate the r and s parameters from simulation data.
To apply the extended Zimm–Bragg theory based on
the helical-block nucleation, we obtain the number of
tries and successes for adding or removing a helical resi-
due from a helical block of length m (nm,100?110,try,
nm,100?110,suc, nm,110?100,try, and nm,110?100,suc). If the
nucleation block size is n, then
For s, we simply collect the statistics for m ? n, regard-
less the length of helix block, in order to increase the sta-
tistical certainty of the result.
Reduced specific heat
The specific heat (CV) or reduced specific heat51(cV*)
is calculated from the energy fluctuation theorem
¼hE2i ? hEi2
where kB, T, and E denote the Boltzmann constant, abso-
lute temperature, and the total energy of the system,
respectively,and ‘‘<>’’ means
sampled conformations. A reduced, dimensionless tem-
perature T*(:kBT/eHB) will be used in this paper.
The model is simulated at 19 different temperatures
(1/T* ¼ 0, 0.415, 0.830, 1.245, 1.660, 2.075, 2.490, 2.905,
3.320, 3.735, 4.150, 4.565, 4.980, 5.395, 5.810, 6.225,
6.640, 7.055, and 7.470). Each simulation starts from an
equilibrated conformation from a previous simulation at
a higher temperature and the temperature is gradually
decreased from the highest temperate to the desired tem-
perature. Each temperature is independently simulated 20
times for error estimation. For each simulation, the first
half of the simulation is for equilibration and the second
half is for data collection. The total number of MC steps
for each simulation is between 2 millions for the highest
temperature and 38 millions for the lowest temperature.
The program is written in FORTRAN. For N ¼ 32, it
takes about 90 h for all of the 19 simulations on a PC
with 2.4 GHz Xeon CPU.
While the majority of our simulations are on the chain
length of 32 residues, we also simulate chain length of
12, 16, 24, 50, 65, 80, and 100. The sizes of simulation
boxes are adjusted according to chain size to avoid self-
interaction due to the period boundary condition. For
example, the size of the simulation box is 70 3 70 3 70
for a chain of 100 residues. Simulating chains with differ-
ent lengths allows us to examine how chain length affects
the performance of the Zimm–Bragg theory.
The reduced specific heat as a function of inverse tem-
perature is shown in Figure 3. There is a peak at 1/T* ¼
4.15 indicating a thermodynamic transition for the simu-
lated single homopolypeptide chain. Examination of
structures reveals that this process is a transition from
coil to helix.
Nucleation and propagation parameters
for the lattice model
The nucleation and propagation parameters (r, s) are
calculated from equilibrium constants from the simula-
tions. Figure 4 shows the values of r and s that are cal-
culated either from K100$110 and K000$010 or from
K101$111and K000$010at different temperatures. The fig-
ure shows that two parameters, r and s, have very differ-
ent temperature dependence. The nucleation parameter
varies between 0 and 0.4 over the entire temperature
range simulated while the propagation parameter changes
three orders of magnitude in the same temperature
range. This means that the initiation of the first helical
residue is only slightly easier (high r values) at high
temperatures than at low temperature. Once the first hel-
ical residue is formed, helical propagation is much easier
at lower temperatures when helix structures are more
The Helix–Coil Transition Revisited
stable. Similar temperature dependences of r and s are
also observed by other theoretical studies.32,51
Figure 4 also shows that the differences between the
two sets of r and s are small, especially at low tempera-
tures. This validates the approaches we used to calculate
r and s. Moreover, the results further suggest that helical
nucleation and propagation parameters provide a reason-
able description for the formation and breakage of helical
blocks (K101$111) as well. We used r and s calculated
from K100$110and K000$010in the remaining sections of
this article, because this calculation is easier and concep-
tually more direct.
Helical ratio and helical block length
The average helical ratio (helicity) and the average
length of helical blocks can be obtained directly from
simulations. They can also be calculated from the original
Zimm–Bragg theory, the large-N approximation, and the
large-k approximation by using r and s derived from the
simulation data as mentioned above.
As shown in Figure 5(a), the helical ratios determined
from Eq. (8) (the original Zimm–Bragg theory) and
Eq. (11) (the large-N approximation) are in a good agree-
ment with the direct simulation results in the entire tem-
perature range simulated. There are also good agreements
between the simulated mean lengths of a-helical blocks
and calculated values based on the original Zimm–Bragg
theory [Fig. 5(b)]. The large-N approximation, however,
produces unphysically high lengths for helical blocks at
low temperatures (much greater than the maximum pos-
sible length of 28 residues). The results from the large-k
Reduced specific heat cV* as a function of reduced inverse temperature 1/T*.
Insets are the typical snapshots of the configurations at associated reduced
inverse temperature. The error bars indicate the standard deviations of all
examined trajectories, and are shown unless smaller than the sizes of underlying
Nucleation parameter r and propagation parameter s obtained from the
equilibrium simulations. They are calculated from the combination of two
equilibrium constants (either from K100$110and K000$010or from K101$111
The helical ratio y (a) and the averaged length of a-helical blocks, L (b)
obtained directly from simulation (solid circles) or calculated from the Zimm–
Bragg theory using the original equation [Eqs. (8) and (10), diamonds], large-N
approximation [Eqs. (11) and (12), open triangles], and large-k approximation
[Eqs. (11) and (16), open circles].
Y. Chen et al.
approximation make a significant improvement over
those from the large-N approximation by satisfying the
low temperature high-helicity limit.
One interesting question is whether or not the agree-
ment between the Zimm–Bragg theory and the large-N
approximation will improve as the chain length increases.
Figure 6 compares the average lengths of helical blocks
for different chain lengths at a temperature that is lower
than the transition temperature. Indeed, the results from
the large-N approximation become in close agreement
with those from the original Zimm–Bragg theory at N ¼
100. However, we surprisingly found that the agreement
between simulation results and the results from the
Zimm–Bragg theory (no matter from the original formu-
lae, large-N approximation, or large-k approximation)
was not satisfactory even when chain
Nucleation by short helical block
The significant difference between what predicted from
the Zimm–Bragg theory and what obtained from simula-
tions in Figure 6 implies that there is room for improve-
ment of the original Zimm–Bragg model.
In the Zimm–Bragg theory, it is assumed that the
propagation parameter (s) is independent of the length
of a-helical block. That is, the equilibrium constant
Km,100$110between adding a helical residue to a helical
block of length m and its reverse process is a constant
(equal to s), independent of m.
In Figure 7, Km,100$110is plotted as a function of m
for the homopolypeptide chain of 32 residues. The
results show that Km,100$110could be regarded as a con-
stant only after helical block is longer than six residues
for the model studied here. K1,100$110 is the smallest
among Km,100$110.This means that the formation of the
first helical residue is the most difficult. K2,100$110is the
second smallest among Km,100$110.So, the second helical
residue is still difficult to form. Thus, while it is mostly
true that the propagation parameter is the same after
nucleation it takes more than one residue for the com-
pletion of nucleation process.
To improve the Zimm–Bragg model, we assume that
helix propagation is initiated by a helical block of length
n rather than a single helical residue. The block nuclea-
tion parameter can be calculated from Km,100$110, m ? n
(see methods). The new block nucleation parameter and
propagation parameter are then used to calculate helical
ratio and average helical-block length. Figure 8 shows
that when helical blocks are used for nucleation, the
agreement between predicted and simulated values for
average helix length and helical ratio are significantly
improved, at long chain length, in particular. A block of
two residues makes the largest improvement in agree-
ment between theory and simulation results.
As in Figure 5 but as a function of chain length. The simulation temperature is
at Tc*/1.3. Tc*refers to the temperature at which specific heat is at its
maximum. Simulated chain lengths are 12, 16, 24, 32, 50, 65, 80, and 100.
Equilibrium constants Km,100$110as a function of the existing helical block
length m. The lines are for guiding eyes.
The Helix–Coil Transition Revisited
In this article, we investigate the helix–coil transition
of a bond-fluctuation lattice model by using MC simula-
tion techniques. For a chain length of 32 residues, the
simulation result can be described reasonably well by the
Zimm–Bragg theory. This is significant because our lat-
tice model allows the formation of nonnative H-bonds.
The result is consistent with other simulation stud-
ies.32,38For example, Takano et al.38showed a qualita-
tive agreement between the Zimm–Bragg theory and the
all-atom molecular dynamics simulation of a short polya-
lanine chain (N ¼ 15).
The commonly-used large-N approximation for the
Zimm–Bragg theory is shown not accurate for predicting
the average helix length based on nucleation and propa-
gation parameters. This is because a chain length of 32
residues is not long enough to satisfy the large-N approx-
imation. Similar result has been observed by Ohkubo
and Brooks.32In fact, our detailed analysis indicates that
the large-N approximation would be satisfied only if
N ? s/r rather than N ? 1. Because the helical lengths in
proteins are often short, it is necessary to have a simple
equation that is applicable at the medium chain length as
well. We proposed a large-k approximation for the aver-
age helix length. The result of the new equation is in
excellent agreement with the result from the more
sophisticate equation in the original Zimm–Bragg theory.
The large-k approximation allows a simple way to relate
the helical ratio and the average helix length to the
nucleation and propagation parameters in accuracy simi-
lar to the original Zimm–Bragg theory.
We further show that the propagating ability is nearly
independent of the length of a-helical block if the block
is longer than a certain length. This validates the basic
assumption in the Zimm–Bragg theory that the propaga-
tion parameter is constant after the nucleation process.15
The phenomena is reminiscent of the radical polymeriza-
tion in polymer chemistry, in which the polymerizing
ability of each step is nearly independent of the chain
length after initiation.68However, the length of nuclea-
tion block is greater than 1. The formations of first sev-
eral helical residues (not just the first residue) are all dif-
ficult although the first helical residue is the hardest to
form. This is understandable because the a-helical H-
bond happens not between two neighboring residues but
between two residues that has a sequence separation of
Our simulation discovers that the single-residue nucle-
ation used in the Zimm–Bragg theory leads to a system-
atic deviation from the simulation results as the chain
length becomes longer. We demonstrate that this devia-
tion can be removed by using a mechanism of helix-
We thank Professor Jian Zi for a critical reading of our
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2. Pauling L, Corey RB, Barnson HR. The structure of proteins: two
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