THE JOURNAL OF CHEMICAL PHYSICS 129, 225101 共2008兲
A statistical mechanical model for antiparallel ␤-sheet/coil equilibrium
Liu Honga兲
Zhou Pei-Yuan Center for Applied Mathematics, Tsinghua University,
Beijing 100084, People’s Republic of China
共Received 24 July 2008; accepted 24 October 2008; published online 9 December 2008兲
In this paper, a simple statistical mechanical model for the antiparallel ␤-sheet/coil equilibrium is
constructed. It is based on the transfer matrix method, which has been widely used in the helix/coil
transition theory. However, to correctly represent the structure of antiparallel ␤-sheet, we have to
extend the former numerical matrix to an 11⫻ 11 operator one. All calculation rules of the operators
are given, which can be automatically done by computers. In the end, the partition functions for
homopolymers of varied lengths, applications to experimental data, and detailed antiparallel
␤-sheet/coil transition are shown. © 2008 American Institute of Physics. 关DOI: 10.1063/1.3028635兴
I. INTRODUCTION
Peptides that fold into ␣-helices and ␤-sheets in isolation
have long been an interesting system for the study of protein’s secondary structure.1–3 To qualitatively interpret observed experimental results, a statistical mechanical model
that considers every conformation adopted by the peptide is
essential.
In the late 1950s, Zimm and Bragg4 and Lifson and
Roig5 proposed their classical models for helix/coil transition, respectively. After that, most works focused on how to
include more factors that may influence the stability of a
helix. Doig and co-workers6–8 and Andersen and Tong9
discussed preferences for the N and C cappings. Cochran
et al.10,11 and Petukhov et al.12,13 added N1, N2, and N3
preferences. Harper and Rose14 studied capping motifs.
Scholtz et al.,15 Shalongo and Stellwagen,16 and Stapley et
al.17 extended the Lifson–Roig 共LR兲 model to include helix
dipoles and side chain interactions, which were also proposed for the Zimm–Bragg 共ZB兲 model by Vasquez and
Scheraga,18 Roberts,19 and Gans et al.20 Adapting the LR
model to different kinds of helices, ␣-, ␲-, and 310-helices,
was discussed by Rohl and Doig.21
Meanwhile, far less works have been done on ␤ structures. Birstein and Ptitsyn22 and David and co-workers23–25
studied helix/sheet transition caused by extension. Munoz
et al.,26,27 Klimov and Thirumalai,28 Dinner et al.,29 and
Chen and Dill30,31 proposed thermodynamical models for
␤-hairpin. Mattice and Scheraga32–35 and Sun and Doig36
used the transfer matrix method to describe antiparallel
␤-sheet. However, the size of their constructed matrix grows
rapidly with the increase in ␤-strands, which makes their
method hard to be applied to long peptides. In the presence
of ␤-sheets, theoretical modeling is far more complex than
that for ␣-helix solely. This is mainly due to the inherent
difficulties caused by sequentially distant interactions in the
␤-sheet1–3, as residues in the sheet may form hydrogen bonds
a兲
Electronic mail: [email protected].
0021-9606/2008/129共22兲/225101/7/$23.00
with some others separated by a long sequence of amino
acids rather than the local interactions between i , i + 4 residues in the ␣-helix.
In this paper, we will develop a novel statistical mechanical model for the antiparallel ␤-sheet/coil equilibrium,
which can generate all possible conformations with variable
lengths of ␤-strands linked by ␤-turns. Our model is based
on the transfer matrix method, which was first proposed in
the ZB model and then widely used in later various helix/coil
transition theories. Although it will be an 11⫻ 11 operator
matrix rather than the former numerical one, all calculation
rules of the operators are known and can be automatically
done by computers. In the following, our discussions will be
limited to homopolymers for simplicity. However, this does
not affect the generality of our model and can be easily extended to heteropolymers as shown in Sec. II D.
II. ANTIPARALLEL ␤-SHEET MODEL
A. Possible conformations of antiparallel ␤-sheet
In this section, we try to describe all chemically allowed
conformations of the antiparallel ␤-sheet, which depend on
the state of each residue. For the residues in ␤-strands, their
angle pairs 共␾ , ␺兲 take values in the upper left quadrant of
the Ramachandran plot.1,5 They can form two kinds of
matching pairs, which we call “column” in Fig. 1: one with
two residues and the other with more than three residues
共including three兲, joined by noncovalent bonds.36 The residues in columns can be further classified into two types. The
first type forms matching pairs with other residues on both
sides and will be recorded as di, while the second type can
only form matching pairs on one side, and thus will be noted
as ai or bi according to their positions in the columns. The
upper one is ai, and the nether one is bi. Besides, any residue
in the ␤-turns is assigned state ti, and those do not belong to
␤-strands or ␤-turns will all be regarded as coil residues 共c兲.
Here we have to distinguish the odd and even ␤-strands or
the odd and even ␤-turns according to their arranging order
in each ␤-sheet. i = 1 for odd ones; i = 2 for even ones. The
above notations are summarized in Table I.
129, 225101-1
© 2008 American Institute of Physics
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225101-2
H
H
O
H
O
O
H
H
O
O
H
H
O
c
O
a
b
t
b2
c
H
t
a
t1
N
t
t
d
b
N
N
H
N
O
O
O
H
O
O
H
O
H
O
H
a
O
N
H
N
N
O
(c)
t
d
t
t
t
d
b
t
t
d
N
N
N
N
(b)
N
H
H
O
(a)
N
N
H
H
N
N
O
(a)
a1
NH3+
N
N
O
O
N
N
N
H
J. Chem. Phys. 129, 225101 共2008兲
Liu Hong
(d)
H
(e)
(f)
O
FIG. 2. 共Color online兲 Possible conformations of a given ␤-strand in the
antiparallel ␤-sheet. Odd and even ␤-strands 共␤-turns兲 are not distinguished
for simplicity.
-
O2C
H
column
the residues one to one. Their calculation rules guarantee that
all generated conformations are chemically allowed and will
be given in Sec. II C.
B. Operator transfer matrix for antiparallel ␤-sheet
(b)
FIG. 1. 共a兲 Structure of antiparallel ␤-sheet. 共b兲 Corresponding states of
residues and columns.
The statistical weights for each state of residues are also
given in Table I. Value 1 is arbitrarily assigned to coil residues, since only the relative ratio is effective. Factor ␤ is
given to residues in the ␤-strands, representing the stabilizing effect within the matching pairs. Factor t stands for the
torsion angle constraint of the residues in the ␤-turns.
However, since the original transfer matrix method cannot guarantee the correct generation of antiparallel ␤-sheets
automatically as that in ␣-helix, we need to modify the
former numerical matrix into an operator one. The
operators—the capitals in Table I, correspond to the states of
The conformation of any given ␤-strand depends on the
states of the residues in its two neighboring strands, as what
we can see from the columns in Fig. 1. Except for the beginning and ending ␤-strand of a ␤-sheet, they depend on only
one. Through a systematical exploration, we can easily show
all chemically allowed conformations that a given ␤-strand
can adopt in Fig. 2.
When there are only two ␤-strands 共␤-hairpin兲, the
possible conformations are all in the same form:
c ¯ ca1 ¯ a1t1 ¯ t1b2 ¯ b2c ¯ c, which can be shortened as
ca1t1b2c as in Fig. 2共a兲. When there are more than two
␤-strands, the possible conformations of the central ␤-strand
can only be one of the five following forms: biai 共short
for bi ¯ biai ¯ ai兲, bidiai 共bi ¯ bidi ¯ diai ¯ ai兲, diai
共di ¯ diai ¯ ai兲, bidi 共bi ¯ bidi ¯ di兲, and di 共di ¯ di兲 共i = 1 or
2兲, as shown in Figs. 2共b兲–2共f兲, respectively. Taking in the
residues in random coils and ␤-turns, we can list all possibilities of the state transfer along the chain in Table II. “State
transfer” means as follows: if the state of one residue is
TABLE I. States and statistical weights of residues in antiparallel ␤-sheet.
State
Weight
Operator
Description
c
1
C
a1
␤
A1
b1
␤
B1
d1
t1
␤
t
D1
T1
a2
␤
A2
b2
␤
B2
d2
t2
␤
t
D2
T2
Coil residue
Residue in odd ␤-strand, upper in column, forming
matching pairs on one side
Residue in odd ␤-strand, nether in column, forming
matching pairs on one side
Residue in odd ␤-strand, middle in column, forming
matching pairs on both sides
Residue in odd ␤-turn
Residue in even ␤-strand, upper in column, forming
matching pairs on one side
Residue in even ␤-strand, nether in column, forming
matching pairs on one side
Residue in even ␤-strand, middle in column, forming
matching pairs on both sides
Residue in even ␤-turn
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Antiparallel ␤-sheet/coil equilibrium
225101-3
J. Chem. Phys. 129, 225101 共2008兲
given, then which state can the next residue adopt?
The state transfer rules in Table II are the foundation in
constructing our operator transfer matrix, just like that in the
ZB and LR models.4,5 In general, the vertical column of the
matrix corresponds to the state of residue i, while the horizontal column corresponds to the state of residue i + 1. Thus
冢
c
c
b1
d1
C
0
0
b1 C
d1 0
a1 0
t01
0
t1
0
b2 C
d2 0
a2 0
t02
0
t2
0
␤B1 ␤D1
0
␤D1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
␭␤B1 ␭␤D1
if the state of residue i + 1 is not allowed to follow that of
residue i by the above transfer rules, then their corresponding
value in the matrix will be zero; otherwise the value will be
the product of the statistical weight and the operator of residue i + 1 in Table I. So we can construct the operator transfer
matrix as
t01
t1
b2
d2
a2
t02
t2
␩␭␤A1 0
0
␤A1
␤A1 tT1
␤A1 tT1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
a1
0
0
tT1
0
0
tT1 ␭␤B2 ␭␤D2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
␤B2
0
0
0
0
␤D2
␤D2
0
0
0
␤A2 0 0
␤A2 tT2 0
␤A2 tT2 0
0
0 tT2
0
0 tT2
冣
.
共1兲
In the above matrix, t01 and t02 stand for residues in odd and even ␤-turns, too. They are added to make sure that there are
at least two residues in each ␤-turn. In natural proteins, ␤-turns with two to six residues are all widely seen and nearly 70%
␤-hairpins with loop length ⱕ7.37 However, here we set no upper bound for the length of ␤-turns. Besides, to account for the
initiation effect, two additional statistical weights are introduced: ␩ for the initiation of a ␤-sheet and ␭ for the ␤-strand. Then
its submatrix
M=
冢
C
C
0
0
0
0
C
0
0
0
0
0
0
␤B1 ␤D1
0
␤D1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
␭␤B1 ␭␤D1
␩␭␤A1 0
0
␤A1
␤A1 tT1
␤A1 tT1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
tT1
0
0
0
0
0
0
0
tT1 ␭␤B2 ␭␤D2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
␤B2
0
0
0
0
␤D2
␤D2
0
0
0
is what we want. Now the partition function for a chain with
N residues is
Z = C0VM NU兩C0=1 ,
where
V = 共1 0 0 0 0 0 0 0 0 0 0 兲
is a 1 ⫻ 11 vector and
共3兲
␤A2 0 0
␤A2 tT2 0
␤A2 tT2 0
0
0 tT2
0
0 tT2
冣
共2兲
U = 共1 1 0 0 0 0 1 0 0 0 0 兲T
is an 11⫻ 1 vector. Here we add an additional operator C0 to
facilitate the calculation, which means the beginning of a
chain.
Based on the partition function in Eq. 共3兲, we can easily
predict some basic statistical properties of the antiparallel
␤-sheet/coil equilibrium, such as the average number of resi-
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225101-4
J. Chem. Phys. 129, 225101 共2008兲
Liu Hong
TABLE II. State transfer rules for antiparallel ␤-sheet.
Column
1
2
3
4
5
6
7
8
9
10
11
12
13
State transfer
Figure
ca1t1
t 2b 1a 1t 1
t 2b 1d 1a 1t 1
t 2d 1a 1t 1
t 2b 1d 1t 1
t 2d 1t 1
t 1b 2a 2t 2
t 1b 2d 2a 2t 2
t 1d 2a 2t 2
t 1b 2d 2t 2
t 1d 2t 2
t 2b 1c
t 1b 2c
2共a兲
2共b兲
2共c兲
2共b兲
2共e兲
2共f兲
2共b兲
2共c兲
2共d兲
2共e兲
2共f兲
2共a兲
2共a兲
Description
␤-sheet begins with a1.
Odd ␤-strands begin with b1 or d1; and end at d1 or a1.
b1 appears before d1 and a1, after t2; d1 appears before
a1 and t1, after t2 and b1; a1 appears before t1, after b1
and d1.
Even ␤-strands begin with b2 or d2; and end at d2 or a2.
b2 appears before d2 and a2, after t1; d2 appears before
a2 and t2, after t1 and b2; a2 appears before t2, after b2
and d2.
␤-sheet ends at b1 or b2.
dues in ␤-strands 共具nb典兲, the average number of residues in
␤-turns 共具nt典兲, the average number of ␤-strands 共具lb典兲 and
␤-sheets 共具kb典兲, etc.,
具nb典 =
具nt典 =
具lb典 =
⳵ ln Z
,
⳵ ln ␤
共4兲
共12兲
A2T2C = A2T2A1 = A2T2T1 = 0,
A 1T 2A 1 = A 1A 1,
A 1T 2T 1 = A 1T 1,
共13兲
⳵ ln Z
,
⳵ ln t
共5兲
⳵ ln Z
,
⳵ ln ␭
共6兲
A1T2B1 = A1T2D1 = A1T2C = 0,
C 0T 2C = C 0,
⳵ ln Z
.
具kb典 =
⳵ ln ␩
共7兲
C. Calculation rules of operators
Now, the major question is how to calculate the operators. We argue that the parameters 共␩ , ␭ , ␤ , t兲 can communicate with the operators freely, while the operators
共A1 , A2 , B1 , B2 , D1 , D2 , T1 , T2 , C , C0兲 cannot communicate
with each other and must be calculated from left to right
according to the following rules:
T 1T 1 = T 1,
m+1 n−1
n
C 0A m
1 A 2T 2D 1 = C 0A 1 A 2 T 2,
A 2T 2B 1 = T 2,
T 2T 2 = T 2,
C 0C = C 0,
ⴱ 0 = 0 ⴱ = 0. 共8兲
For odd ␤-turns:
m+1 n−1
n
C 0A m
2 A 1T 1D 2 = C 0A 2 A 1 T 1,
A 1T 1B 2 = T 1,
共9兲
A1T1C = A1T1A2 = A1T1T2 = 0,
A 2T 1A 2 = A 2A 2,
A 2T 1T 2 = A 2T 2,
共10兲
C 0T 2A 1 = C 0A 1,
共14兲
C0T2B1 = C0T2D1 = C0T2T1 = 0.
The above rules can be deduced from the structural consideration of the antiparallel ␤-sheet. For example, in Eq. 共9兲,
as no conformations like a1t1c, a1t1a2, and a1t1t2 are chemically allowed, we have A1T1C = A1T1A2 = A1T1T2 = 0. Residues in states a1 and b2 of two successive ␤-strands can form
a matching pair; thus we can simplify A1T1B2 = T1. The orim+1 n−1
n
gin of the rule C0Am
2 A1T1D2 = C0A2 A1 T1 is more complex. As we have pointed out in Sec. II A, a residue in state
d2 must form a matching pair with another two residues on
both sides. So when it forms a matching pair with residue a1
in the former ␤-strand on the upper side, we change its state
to a2 in order to make sure that it will form another matching
pair with residues in the latter ␤-strand on the nether side.
All rules in Eqs. 共8兲–共14兲 can be obtained in the same way. In
fact, we can show that these rules are self-consistent and
sufficient for all calculations.
Using rules in Eqs. 共8兲–共14兲, we can simplify any operator chains obtained by multiplying transfer matrices in the
calculation of the partition function in Eq. 共3兲. If all operators in a chain can be canceled at last, except the parameters
共␩ , ␭ , ␤ , t兲, its corresponding statistical weight will appear in
the final partition function; otherwise its contribution will be
zero.
A2T1B2 = A2T1D2 = A2T1C = 0,
D. Extension to heteropolymers
C 0T 1C = C 0,
Extending our above model to heteropolymers is quite
easy. We only need to correlate the statistical weights ␩, ␭,
␤, and t with the amino acid sequence of the chain. As an
example, for Gly, we have ␩G, ␭G, ␤G, and tG. Its corresponding transfer matrix is
C 0T 1A 2 = C 0A 2,
共11兲
C0T1B2 = C0T1D2 = C0T1T2 = 0.
For even ␤-turns:
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Antiparallel ␤-sheet/coil equilibrium
225101-5
MG =
冢
C
C
0
0
0
0
C
0
0
0
0
0
0
␤ GB 1
␤ GD 1
0
␤ GD 1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
␭ G␤ GB 1 ␭ G␤ GD 1
J. Chem. Phys. 129, 225101 共2008兲
␩ G␭ G␤ GA 1 0
0
␤ GA 1
␤ GA 1
t GT 1
␤ GA 1
t GT 1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
t GT 1
0
0
0
0
0
0
0
t GT 1 ␭ G␤ GB 2 ␭ G␤ GD 2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Thus, the partition function for a six-residue protein chain
AGGPGA will be Z = C0VM AM GM GM PM GM AU 兩C0=1. The
operators and their calculation rules remain the same.
␤ GB 2
0
0
0
0
␤ GD 2
␤ GD 2
0
0
0
0
␤ GA 2 0
␤ GA 2 t GT 2 0
␤ GA 2 t GT 2 0
0
0 t GT 2
0
0 t GT 2
冣
.
Z 5 = 1 + 2 ␩ ␭ 2␤ 2t 2 + ␩ ␭ 2␤ 2t 3 ,
共15兲
共18兲
Z 6 = 1 + 3 ␩ ␭ 2␤ 2t 2 + 2 ␩ ␭ 2␤ 2t 3 + ␩ ␭ 2␤ 2t 4 + ␩ ␭ 2␤ 4t 2 ,
共19兲
E. Comparison with previous ␤-sheet/coil models
Munoz et al.,26,27 Klimov and Thirumalai,28 Dinner et
al., and Chen and Dill30,31 proposed thermodynamical
models for the ␤-sheet/coil equilibrium. However, all these
models are limited to ␤-hairpins and cannot be extended to
more complicated antiparallel ␤-sheet.
Mattice and Scheraga32–35 developed transfer matrix
based models for antiparallel ␤-sheet conformations. They
included parameters for residues in turns, interior ␤-sheets
and the first ␤-strand, the same as ours. However, the size of
their matrix is given by I共I + 3兲 / 2, where I is the maximum
number of the residues allowed in a ␤-strand. This made the
construction of their matrix rather hard and also greatly restricted the application of their method to longer polymer
chains.
Sun and Doig36 constructed their model on columns,
which are residues joined by noncovalent bonds within the
␤-sheet, rather than single amino acid. This clever treatment
largely reduced the size of their matrix but could not avoid
the intrinsical difficulty. When the number of ␤-strands increases from 2 to 4, their matrix size grows from 5 to 38.
Besides, their method is only valid for homopolymers.
29
III. SIMULATION RESULTS
A. Sample partition functions for homopolymers
Z 7 = 1 + 4 ␩ ␭ 2␤ 2t 2 + 3 ␩ ␭ 2␤ 2t 3 + 2 ␩ ␭ 2␤ 2t 4 + 2 ␩ ␭ 2␤ 4t 2
+ ␩ ␭ 3␤ 3t 4 + ␩ ␭ 2␤ 2t 5 + ␩ ␭ 2␤ 4t 3 ,
Z 8 = 1 + 5 ␩ ␭ 2␤ 2t 2 + 4 ␩ ␭ 2␤ 2t 3 + 3 ␩ ␭ 2␤ 2t 4 + 3 ␩ ␭ 2␤ 4t 2
+ 2 ␩ ␭ 3␤ 3t 4 + 2 ␩ ␭ 2␤ 2t 5 + 2 ␩ ␭ 2␤ 4t 3 + 2 ␩ ␭ 3␤ 3t 5
+ ␩ ␭ 3␤ 4t 4 + ␩ ␭ 2␤ 2t 6 + ␩ ␭ 2␤ 4t 4 + ␩ ␭ 2␤ 6t 2 .
Z1 = Z2 = Z3 = 1,
共16兲
Z 4 = 1 + ␩ ␭ 2␤ 2t 2 ,
共17兲
共21兲
B. Application to experimental data
Sharman and Searle38 synthesized a 24-residue peptide
that folds into a ␤-meander and a shorter peptide with 9–24
residues that folds into a ␤-hairpin. The meander is 50%
folded, while the hairpin is 20% folded in methanol at 298 K.
Our current model can be applied to the above experimental data to estimate the parameters. Since a 16-residue
hairpin has at most 14 residues in the ␤-strands 共as 2 in the
␤-turn兲, the sheet content is given by the average number of
residues in the ␤-strands divided by 14. According to Sun
and Doig,36 we have
⳵ ln Z共16兲兩lb=2
Based on our above statistical mechanical model for the
antiparallel ␤-sheet, we can easily get the partition function
for peptides of any length 关Eq. 共3兲兴. Here, we list some results for homopolymers with one to eight residues separately
and their corresponding conformations in Fig. 3. The partition function for longer homopolymers or heteropolymers
can also be obtained automatically by computers.
共20兲
⳵ ln ␤
冒
14 = 0.2,
共22兲
where Z共16兲 兩lb=2 is the partition function for a 16-residue
␤-hairpin. Similarly, a 24-residue ␤-meander has at most 20
residues in the ␤-strands; thus
⳵ ln Z共24兲兩lbⱕ3
⳵ ln ␤
冒
20 = 0.5,
共23兲
where Z共24兲 兩lbⱕ3 is the partition function for a 24-residue
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J. Chem. Phys. 129, 225101 共2008兲
Liu Hong
N=4
N=8
c
t1
c
a1
ηλ β t
2 22
1
c
c
c
c
N=5
b2
t1
c
a1
+
ηλ2β2t3 t1
c
c
t1
a1 c
t1
b2
c
c
t1
b2
t1
a1
t1
b2 c
t1
a1 c
+
2ηλ2β2t2
c
c
c
c
t1
a1 c
t1
b2
t1
a1 c
c
c
c
18
c
16
5ηλ β t
2 22
1
t1
1
c
c
c
c
c
t1
a1 c
c
c
14
12
< nb>
225101-6
+
t1
b2 c
t1
b2 c
t1
a1 c
t1
a1
t1
b2 c
t1
b2 c
8
+
+
c
c
c
c
c
t1
a1 c
c
c
t1
t1
b2
a1
6
4
N=6
t1
c
c
t1
a1 c
4ηλ2β2t3 t1
c
c
+
t1
b2 c
t1
a1
t1
t1
+
+
t1
t1
b2
t1
2ηλ2β2t3 t1
+
t1
a1 a1
t1
b2 b2
c
c
ηλ2β4t2
b2 c
a1 c
t1
t1
t1
b2 c
c
c
c
a1
b2 c
c
t1
t1
a1
t1
t1
b2 c
c
t1
a1 a1
c
c
t1
b2 b2
c
t1
b2 c
t1
t1
b2
t1
t1
a1
t1
t1
a1
t1
t1
b2
t1
t1
b2 c
t1
a1 c
t1
a1 a1
c
+
c
c
c
c
t1
+
c
a1 c
+
t1
t1
t1
b2
t1
a1
t1
t1
b2
t1
a1 c
c
b2 c
a1
t1
t1
b2 c
t1
a1 c
t1
b2 c
t1
b2
a1 c
t1
a1
c
t1
b2 c
t1
b2 c
c
t1
a1 a1
t1
b2 b2
c
t1
a1 a1
t1
b2 b2 c
+
t1
t1
t1 a1
t1
t1 b2
ηλ2β2t5 t1
t1
t1
a1
t1
d2
b1
b2 b2
c
t2
d2
t1
b1
a1
t1
b2 b2
t1
t1
t1
2ηλ3β3t5
b2 b2 c
t1 t1 a1
t1
t1
t1 b2 b2
a1 c
t1
t1
+
t2
t2 ηλ2β2t6
t2
a1
a1 a1
t1 b2 c
a1 a1
(b)
t1
t1 b2 b2 c
a1
t1
d2
c
b1
t1 t1
t1
t1
t1
d2
t1
b1
t2
t2
+
β
5
t2
t1
4
a1
b2
t1
a1 a1 a1
t1
t1
b2 b2 b2
a1
t1
d2
3
2
1
t2
b1
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
t2
ηλ2β6t2
t1
4
2
t1
t1
b2 b2
a1
6
t1
+
2ηλ3β3t4 t
1
ηλ3β4t4
a1 a1
t1 b2
a1 a1 c
ηλ2β4t4
a1 a1
ηλ2β4t3 t1
t1
c
+
c
t1
t1
2ηλ2β4t3 t1
t1
+
c
+
c
+
4ηλ2β2t2
ηλ3β3t4
t1
8
c
t1
a1 c
t1
2ηλ2β2t5 t1
+
t1
+
b2 b2 c
t1
t1
t1
c
2ηλ2β2t4
2ηλ2β4t2
t1
t1 b2 c
t1 a1
c
10
3ηλ2β4t2
+
3ηλ2β2t3 t1
β
+
N=7
1
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
(a)
t1
t1
3ηλ2β2t4
ηλ2β2t4
2
t1
+
t1
c
a1
t1
b2
t1
b2 c
t1
a1 c
+
<nt>
3ηλ2β2t2
1
c
a1 c
<lb>
c
λ=1,t=1
λ=1,t=0.5
λ=0.1,t=1
λ=0.1,t=0.5
λ=0.01,t=1
λ=0.01,t=0.5
10
t2
t2
t2
t2
(c)
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
β
1.8
FIG. 3. Possible conformations of the antiparallel ␤-sheet with one to eight
residues, respectively.
1.6
1.4
b
␤-hairpin and a ␤-meander.
In the current case, there is only one ␤-sheet in the
chain, so we can take ␩ = 1. We also assume t = 1. The real
value of t would be smaller than 1. However, considering the
similarity between coil residues and the residues in ␤-turns,
this would not be a very bad approximation. Now we can
determine the values of ␭ and ␤ from Eqs. 共22兲 and 共23兲,
which give ␭ = 0.038 and ␤ = 1.17. Note that here we treat the
24-residue peptide as a homopolymer. So each statistical
weight represents an averaging effect of all amino acids
along the sequence.
The parameters ␭ and ␤ can be compared with analogous parameters for ␣-helix formation—␴ and s in the ZB
model, respectively. In fact, the value of ␤ is quite close to s
共s = 1.14 for Leu兲,39,40 suggesting that the stabilizing effects
between matching residues in ␣-helix and ␤-sheet have almost the same magnitude; while the value of ␭ is a bit larger
than ␴ 共␴ = 0.021 for His兲.39,40 In natural proteins, a ␤-strand
would be interrupted by some residue with a very low
<k >
1.2
1
0.8
0.6
0.4
0.2
(d)
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
β
FIG. 4. 共Color online兲 Statistical properties of antiparallel ␤-sheet/coil equilibrium for a 20-residue homopolymer as a function of ␭, ␤, and t. Here we
choose ␩ = 1. 共a兲 Average number of residues in ␤-strands. 共b兲 Average number of residues in ␤-turns. 共c兲 Average number of ␤-strands. 共d兲 Average
number of antiparallel ␤-sheets. Values of ␭ and t for each curve are listed
on the plot. For red dashed line, ␭ = 1, t = 1; for blue dashed line, ␭ = 1, t
= 0.5; for red solid line, ␭ = 0.1, t = 1; for blue solid line, ␭ = 0.1, t = 0.5; for
red dashed and dotted line, ␭ = 0.01, t = 1; and for blue dashed and dotted
line, ␭ = 0.01, t = 0.5.
probability to adopt the ␤-sheet conformation. This can
largely restrict the average length of ␤-strands. Thus our current study based on the homopolymer model makes an overestimation of ␭.
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225101-7
Antiparallel ␤-sheet/coil equilibrium
C. Antiparallel ␤-sheet/coil transition
Through Eqs. 共4兲–共7兲, we can learn some general properties of antiparallel ␤-sheet/coil equilibrium from the partition function. In Fig. 4共a兲, we see that the antiparallel
␤-sheet/coil transition is a highly cooperative process. With
the increase in the stabilizing effect factor ␤, the average
number of residues in ␤-strands goes larger monotonously.
The sharpness of curves depends on the ␤-sheet initiation
factor ␩, the ␤-strand initiation factor ␭, and the ␤-turn constraint factor t. The smaller ␩, ␭, and t are, the sharper the
transition will be.
The curves in Figs. 4共b兲–4共d兲 are almost in the same
form, which increase quickly at first then drop slowly to
some constant 共nt = 2, lb = 2, kb = 1兲. This phenomenon is a
consequence of the general competition between enthalpy
and entropy effects in the ␤-sheet formation. When the stabilizing effect factor ␤ is small, the chain prefers to form
many short ␤-stands to get larger entropy; when ␤ gets
strong enough, the chain favors the conformation of single
␤-sheet with two long ␤-strands to maximize the number of
residues in ␤-strands.
IV. DISCUSSION
In fact, building an exact statistical mechanical model
for the ␤-sheet formation is a very hard task. This is mainly
due to the nonlocal interactions in the ␤-sheet, which make
the possible conformations of the ␤-sheet too complex to use
the ordinary transfer matrix method. In the case of antiparallel ␤-sheet, we introduce a new operator matrix and some
special calculation rules to guarantee the correct generation
of all chemically allowed conformations. However, for the
parallel ␤-sheet or other mixed types, we must consider the
geometrical constraints on the ␤-turns too. This may cause
great trouble in the theoretical modeling and goes beyond the
scope of all current statistical mechanical models. So how to
take in these geometrical constraints and how to deal with
the multiple connecting patterns of nonantiparallel ␤-sheet
will be central problems in future statistical mechanical modeling for secondary structure formation in peptides.
ACKNOWLEDGMENTS
The author thanks Professor C. C. Lin and Dr. Jinzhi Lei
for their guidance and many useful discussions.
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