# The helix-coil transition revisited.

**ABSTRACT** In this article, we perform a dynamic Monte Carlo simulation study of the helix-coil transition by using a bond-fluctuation lattice model. The results of the simulations are compared with those predicted by the Zimm-Bragg statistical thermodynamic theory with propagation and nucleation parameters determined from simulation data. The Zimm-Bragg theory provides a satisfactory description of the helix-coil transition of a homopolypeptide chain of 32 residues (N = 32). For such a medium-length chain, however, the analytical 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 analytical large-eigenvalue (lambda) 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 transition for longer chains. We assume that the single-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 significantly improved agreement with our simulation data.

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Page 1

proteins

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

INTRODUCTION

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: jdding1@fudan.edu.cn

Received 23 November 2006; Revised 9 February 2007; Accepted 5 March 2007

Published online 27 June 2007 in Wiley InterScience (www.interscience.wiley.com).

DOI: 10.1002/prot.21492

ABSTRACT

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-

tion data.

Proteins 2007; 69:58–68.

V V

C 2007 Wiley-Liss, Inc.

Key words: Zimm–Bragg theory; helix–coil

transition; dynamic Monte Carlo simula-

tion; lattice model; nucleation; protein fold-

ing.

58

PROTEINS

V V

C 2007 WILEY-LISS, INC.

Page 2

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.

Figure 1

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

DOI 10.1002/prot

PROTEINS 59

Page 3

METHODS

The Zimm—Bragg theory

The partition function Q in the Zimm–Bragg theory15

is expressed as53

Q ¼

X

N?4

k¼1

X

minðk;N?k?4Þ

j¼1

Xj;krjsk;

ð1Þ

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,

K1¼ rs;

ð2Þ

ð3Þ

K2¼ K3¼ ??? ¼ Kn¼ ??? ¼ s:

For this simple model, Q has an analytical form given by

Q ¼kN?3

0

ðk0? sÞ ? kN?3

k0? k1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1

ðk1? sÞ

;

ð4Þ

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

residues only.15

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

q

??

=2, k0> k1, k0,

u ¼

1

N ? 4

m ¼@ lnQ

@ lnQ

@ lns;

ð5Þ

@ lnr;

ð6Þ

and

L ¼ ðN ? 4Þu

m¼s

r

@ lnQ=@s

@ lnQ=@r:

ð7Þ

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

u¼

s

N ?4

N?3

k0þ

1

k0?s

??

@k0

@s?

kN?3

1

kN?3

0

1

k0?s

1?k1?s

N?3

k0?s

2

4

þ

k1þ

1

k1?s

??

@k1

@s?

kN?3

0

kN?3

1

1

k1?s

1?k0?s

k1?s

?

@k0

@s?@k1

k0?k1

@s

3

5;

ð8Þ

m¼r

N?3

k0þ

1?k1?s

1

k0?s

kN?3

1

kN?3

0

k0?s

@k0

@rþ

N?3

k1þ

1?k0?s

1

k1?s

kN?3

0

kN?3

1

k1?s

@k1

@r?

@k0

kr?@k1

k0?k1

@r

0

@

1

A;

ð9Þ

L¼s

r

ðN?3

k0þ

1?k1?s

1

k0?sÞ@k0

@s?

kN?3

1

kN?3

0

1

k0?s

kN?3

1

kN?3

0

1

k0?s

k0?s

N?3

þ

ðN?3

k1þ

1?k0?s

1

k1?sÞ@k1

@s?

kN?3

0

kN?3

1

1

k1?s

k1?s

1

k1?s

kN?3

0

kN?3

1

?

@k0

@s?@k1

k0?k1

@s

k0þ

1?k1?s

k0?s

@k0

@rþ

N?3

k1þ

1?k0?s

k1?s

@k1

@r?

@k0

@r?@k1

k0?k1

@r

:

ð10Þ

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

u ¼1

2?

1 ? s

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2s

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð1 ? sÞ2þ 4rs

q

;

ð11Þ

L ¼ 1 þ

1 ? s þð1 ? sÞ2þ 4rs

q

:

ð12Þ

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.

60

PROTEINS

DOI 10.1002/prot

Page 4

expression of Q. Then, Eq. (4) becomes

lnQ ? ðN ? 4Þ lnk0þ lnðk0? sÞ:

ð13Þ

This leads to [cf. Eq. (6)].

m

2rsffiffiffi

D

p

¼

N ? 4

ffiffiffiffi

D

p

þ s þ 1þ

1

ffiffiffiffi

D

p

? s þ 1;

ð14Þ

where

D ? ð1 ? sÞ2þ 4rs:

ð15Þ

Combining Eqs. (11) and (14), we obtain

2

L ¼N ? 4

N ? 3

1 þ

2s ? 2s=ðN ? 3Þ

þ 1 ? s þ 2s=ðN ? 3Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð1 ? sÞ2þ 4rs

q

64

3

75:

ð16Þ

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

approximation k0

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

helicity limit.

N?3? k1

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

K1¼ r1s;

ð17Þ

K2¼ r2s;

Kn¼ rns;

ð18Þ

ð19Þ

ð20Þ

Knþ1¼ Knþ2¼ ??? ¼ s;

where n is the size of the nucleation helix block. The

nucleation parameter rblockfor helical block is given by

rblock¼ r1r2???rn:

ð21Þ

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.

Simulation model

In this

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;

ð22Þ

where

El¼

X

N?1

i¼1

ul;i¼

X

N?1

i¼1

1

2klðli? l0Þ2;

ð23Þ

and

Eu¼

X

N?1

i¼2

uu;i¼

X

N?1

i¼2

1

2kuðcosui? cosu0Þ2:

ð24Þ

The Helix–Coil Transition Revisited

DOI 10.1002/prot

PROTEINS 61

Page 5

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.

ffiffiffip

Ech¼

X

N?3

i¼1

uch;i;

ð25Þ

where

uch;i¼

«ch

0

ðli3 liþ1? liþ2? 0Þ

ðli3 liþ1? liþ2> 0Þ

?

:

ð26Þ

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

EHB¼

X

N?4

i¼1

uHB;i;

ð27Þ

where

uHB;i¼

0

ðhi;j< h1Þ

ðh1? hi;j? h2Þ

ðhi;j< h2Þ

?«HB

0

8

:

<

;

ð28Þ

and

j ? i þ 4:

ð29Þ

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.

ffiffiffiffiffiffiffiffiffi

11:5

p

, h2¼

ffiffiffiffiffiffiffiffiffi

12:5

p

, ech¼

Simulation method

The bond-fluctuation model is simulated with the

same dynamic MC simulation technique employed in

our previous work.51 Every 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

criterion67is satisfied.

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

Figure 2

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.

62

PROTEINS

DOI 10.1002/prot

Page 6

for making a conversion. The probability of propagating

p100?110is thus calculated from

p100!110¼<n100!110;suc>

<n100!110;try>;

ð30Þ

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

rblock¼1

sn

Y

Y

<nm;100!110;suc>=<nm;100!110;try>

<nm;110!100;suc>=<nm;110!100;try>:

n

m¼1

Km;100$110

¼1

sn

n

m¼1

ð31Þ

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

c?

V¼CV

NkB

¼hE2i ? hEi2

NðkBTÞ2;

ð32Þ

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.

theaveraging over

Simulation procedure

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.

RESULTS

Specific heat

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

DOI 10.1002/prot

PROTEINS 63

Page 7

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

Figure 3

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

data points.

Figure 4

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

and K000$010).

Figure 5

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.

64

PROTEINS

DOI 10.1002/prot

Page 8

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 notsatisfactory even when chain

increased.

length was

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.

Figure 6

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.

Figure 7

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

DOI 10.1002/prot

PROTEINS 65

Page 9

DISCUSSION

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

four residues.

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-

block nucleation.

ACKNOWLEDGMENT

We thank Professor Jian Zi for a critical reading of our

manuscript.

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