Shen S., Tuszynski J.A. Theory and Mathematical Methods for Bioformatics

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304 11 Alignment of Primary and Three-Dimensional Structures of Proteins

For D

, the relationship

= −



a,b∈V

p(a)q(b)log

p(a)q(b)

p(a, b)

holds. Moreover, it is clear that D

≤ 0holds.

This shows that D

, and D

are two important indices that can be used

to assess protein similarity. If we denote (A



) as the alignment sequences

for (A, B), then



)=((c

), (c

), ··· , (c

)) .

We denote as p

A,B

(c, d), where c, d ∈ V

, the frequency of (c

)takingthe

value (c, d), and then compute



KL(C|D)=



c,d∈V

C,D

(c, d)k(c, d) ,

HL(C|D)=



c,d∈V

(c)p



(d)k(c, d) ,

(11.4)

where k(c, d) is determined by the data of the PAM or BLOSUM scoring table,

and

(c)=



d∈V

C,D

(c, d) ,p



(d)=



c∈V

C,D

(c, d) .

If KL(A|B) approaches or exceeds D

,(A, B) can be considered similar. If

HL(A|B) is close to or less than D

,(A, B) can be considered not similar.

However, in the sequence alignment, the scores for the segments of the

aligned sequences should be raised in local areas. This will be further explained

later in the book.

Several Typical Scoring Matrices in the Alignment

of Protein Sequences

The following typical scoring matrices in the alignment of protein sequences

are commonly used in various bioinformatics resources and websites. We enu-

merate them as follows:

1. PAM 70 matrix. The following is the PAM 70 matrix [“pam” version 1.0.6

(July 28, 1993)]. The main parameters are

=1.60 ,D

= −2.77 , Max = 13 , Min = −11 ,

where Max and Min are, respectively, the maximum value and the mini-

mum value of the matrix.

The letters are the one-letter codes of the amino acids, “−”isavirtual

symbol. B represents N or D; Z represents Q or E; X represents all kinds

of amino acids.

2. BLOSUM 45 matrix. The data of the statistical samples is obtained

from the following: blocks Database = /data/blocks-5.0/blocks.dat, D

0.3795, D

= −0.2789.

11.1 Structure Analysis for Protein Sequences 305

ARNDCQEGHILKMFPSTWYVBZX-

A 5 -4 -2 -1 -4 -2 -1 0 -4 -2 -4 -4 -3 -6 0 1 1 -9 -5 -1 -1 -1 -2 -11

R -4 8 -3 -6 -5 0 -5 -6 0 -3 -6 2 -2 -7 -2 -1 -4 0 -7 -5 -4 -2 -3 -11

N -2 -3 6 3 -7 -1 0 -1 1 -3 -5 0 -5 -6 -3 1 0 -6 -3 -5 5 -1 -2 -11

D -1 -6 3 6 -9 0 3 -1 -1 -5 -8 -2 -7 -10 -4 -1 -2 -10 -7 -5 5 2 -3 -11

C -4 -5 -7 -9 9 -9 -9 -6 -5 -4 -10 -9 -9 -8 -5 -1 -5 -11 -2 -4 -8 -9 -6 -11

Q -2 0 -1 0 -9 7 2 -4 2 -5 -3 -1 -2 -9 -1 -3 -3 -8 -8 -4 -1 5 -2 -11

E-1-5 0 3-9 2 6-2-2-4-6-2-4-9-3-2-3-11-6-4 2 5-3-11

G 0 -6 -1 -1 -6 -4 -2 6 -6 -6 -7 -5 -6 -7 -3 0 -3 -10 -9 -3 -1 -3 -3 -11

H-4 0 1-1-5 2-2-6 8-6-4-3-6-4-2-3-4-5-1-4 0 1-3-11

I-2-3-3-5-4-5-4-6-6 7 1-4 1 0-5-4-1-9-4 3-4-4-3-11

L -4 -6 -5 -8 -10 -3 -6 -7 -4 1 6 -5 2 -1 -5 -6 -4 -4 -4 0 -6 -4 -4 -11

K-4 2 0-2-9-1-2-5-3-4-5 6 0-9-4-2-1-7-7-6-1-2-3-11

M -3 -2 -5 -7 -9 -2 -4 -6 -6 1 2 0 10 -2 -5 -3 -2 -8 -7 0 -6 -3 -3 -11

F -6 -7 -6 -10 -8 -9 -9 -7 -4 0 -1 -9 -2 8 -7 -4 -6 -2 4 -5 -7 -9 -5 -11

P 0 -2 -3 -4 -5 -1 -3 -3 -2 -5 -5 -4 -5 -7 7 0 -2 -9 -9 -3 -4 -2 -3 -11

S 1 -1 1 -1 -1 -3 -2 0 -3 -4 -6 -2 -3 -4 0 5 2 -3 -5 -3 0 -2 -1 -11

T 1-4 0-2-5-3-3-3-4-1-4-1-2-6-2 2 6-8-4-1-1-3-2-11

W -9 0 -6 -10 -11 -8 -11 -10 -5 -9 -4 -7 -8 -2 -9 -3 -8 13 -3 -10 -7 -10 -7 -11

Y-5-7-3-7-2-8-6-9-1-4-4-7-7 4-9-5-4-3 9-5-4-7-5-11

V -1 -5 -5 -5 -4 -4 -4 -3 -4 3 0 -6 0 -5 -3 -3 -1 -10 -5 6 -5 -4 -2 -11

B -1 -4 5 5 -8 -1 2 -1 0 -4 -6 -1 -6 -7 -4 0 -1 -7 -4 -5 5 1 -2 -11

Z -1 -2 -1 2 -9 5 5 -3 1 -4 -4 -2 -3 -9 -2 -2 -3 -10 -7 -4 1 5 -3 -11

X-2-3-2-3-6-2-3-3-3-3-4-3-3-5-3-1-2-7-5-2-2-3-3-11

- -11 -11 -11 -11 -11 -11 -11 -11 -11 -11 -11 -11 -11 -11 -11 -11 -11 -11 -11 -11 -11 -11 -11 1

ARNDCQEGHILKMFPSTWYVBZX-

A 5-2-1-2-1-1-1 0-2-1-1-1-1-2-1 1 0-2-2 0-1-1 0-5

R -2 7 0 -1 -3 1 0 -2 0 -3 -2 3 -1 -2 -2 -1 -1 -2 -1 -2 -1 0 -1 -5

N-1062-20001-2-30-2-2-210-4-2-340-1-5

D-2-1 2 7-3 0 2-1 0-4-3 0-3-4-1 0-1-4-2-3 5 1-1-5

C-1-3-2-312-3-3-3-3-3-2-3-2-2-4-1-1-5-3-1-2-3-2-5

Q-1100-362-21-2-210-4-10-1-2-1-304-1-5

E-1002-326-20-3-21-2-300-1-3-2-314-1-5

G 0-2 0-1-3-2-2 7-2-4-3-2-2-3-2 0-2-2-3-3-1-2-1-5

H-2010-310-210-3-2-10-2-2-1-2-32-300-1-5

I-1-3-2-4-3-2-3-4-3 5 2-3 2 0-2-2-1-2 0 3-3-3-1-5

L-1-2-3-3-2-2-2-3-2 2 5-3 2 1-3-3-1-2 0 1-3-2-1-5

K-1300-311-2-1-3-35-1-3-1-1-1-2-1-201-1-5

M-1-1-2-3-20-2-2022-160-2-2-1-201-2-1-1-5

F-2-2-2-4-2-4-3-3-201-308-3-2-1130-3-3-1-5

P-1-2-2-1-4-1 0-2-2-2-3-1-2-3 9-1-1-3-3-3-2-1-1-5

S1-110-1000-1-2-3-1-2-2-142-4-2-1000-5

T 0-1 0-1-1-1-1-2-2-1-1-1-1-1-1 2 5-3-1 0 0-1 0-5

W-2-2-4-4-5-2-3-2-3-2-2-2-2 1-3-4-315 3-3-4-2-2-5

Y-2-1-2-2-3-1-2-3200-103-3-2-138-1-2-2-1-5

V 0 -2 -3 -3 -1 -3 -3 -3 -3 3 1 -2 1 0 -3 -1 0 -3 -1 5 -3 -3 -1 -5

B -1 -1 4 5 -2 0 1 -1 0 -3 -3 0 -2 -3 -2 0 0 -4 -2 -3 4 2 -1 -5

Z-1001-344-20-3-21-1-3-10-1-2-2-324-1-5

X 0 -1 -1 -1 -2 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 0 0 -2 -1 -1 -1 -1 -1 -5

--5-5-5-5-5-5-5-5-5-5-5-5-5-5-5-5-5-5-5-5-5-5-5 1

3. BLOSUM 62 matrix. The data of the statistical samples are obtained

from the following: blocks Database = /data/blocks-5.0/blocks.dat, D

0.6979, D

= −0.5209.

4. BLOSUM 80 matrix. The data of the statistical samples are obtained

from the following: blocks Database = /data/blocks-5.0/blocks.dat, D

0.9868, D

= −0.7442.

5. PAM 30 matrix. The following is “pam” version 1.0.6 (28 July 1993). The

main parameters are D

=2.57, D

= −5.06, Max = 13, Min = −17.

306 11 Alignment of Primary and Three-Dimensional Structures of Proteins

11.1.4 Key Points of the Alignment Algorithms

of Protein Sequences

Using the scoring matrix of the protein sequence alignment, the alignment of

protein sequences can be carried out. The dynamic programming algorithm,

SPA (which is based on statistical judgment), and the algorithms for multi-

ple sequence alignment given above, are all based on the data for DNA (or

RNA). Therefore, corresponding complementarity and modiﬁcation should be

made for protein sequences. The main points of modiﬁcation of the related

algorithms are illustrated as follows.

Dynamic Programming Algorithm for the Alignment

of Protein Sequences

The basic steps of the dynamic programming algorithm remain the same. We

compute with the PAM or BLOSUM matrix directly, following the discussion

in Sect. 1.3.

Illustration of the SPA Algorithm

The key point of SPA is the computation of sliding window functions and the

statistical estimations based on them. The sliding window functions are given

by (4.10). They are

w(A, B; i, j, n)=w



[i+1,i+n]

[j+1,j+n]





k=1

w(a

i+k

j+k

) . (11.5)

In the alignment of protein sequences, we only need to change the w(a, b) func-

tion into the corresponding scoring function v(a, b). In the statistical estimate

formulas (4.14) and (4.16) used in SPA, the lack of bias and the consistency of

these estimates can not be ensured, since complete randomness does not hold

in the structure of protein sequences. This shows that SPA can still be used in

the alignment of protein sequences, while more modiﬁcations and computa-

tions are needed to determine the position of the displacement mutation after

the parameter estimation. Moreover, in SPA, further analysis is needed for

the selection of parameters n, h, θ, θ



, τ. The key problems in the alignment

of protein sequences are: how to make use of the probability model and the

limit properties of the stochastic sequence, and how to combine the origin

of the PAM and BLOSUM scoring matrix series to achieve more systematic

results.

Illustration of the Algorithm of the Alignment of Multiple Protein

Sequences

The theory of sequence segments and module structures has nothing to do

with the value of q in V

, thus the deﬁnition of the uniform alignment and the

11.2 Alignment of Protein Three-Dimensional Structures 307

algorithm for multiple sequence alignment can also be used in the alignment

of multiple protein sequences.

Since the multiple sequence alignment given in this book is based on pair-

wise sequence alignment, and protein sequences are relatively short, the dy-

namic programming of their pairwise alignment is comparatively mature, and

many websites and software packages can be used directly. Therefore, the

alignment of multiple protein sequences can be implemented. First we get the

pairwise sequence alignment by the dynamic programming algorithm, and

then the sequence matrix and the module structure matrix of their pairwise

sequence alignment are obtained. Finally, the alignment of multiple protein

sequences can be implemented through the use of Steps 5.3.2 and 5.3.4 in the

algorithm for multiple sequence alignment.

Further discussions are needed concerning whether the theory of the

stochastic model of DNA sequences and performance analysis (such as the the-

ory of complexity analysis and error analysis) can be used in the alignment

and analysis of protein sequences. Some assumptions such as the complete

randomness of the sequences, the limit properties of pairwise independent se-

quences, the characteristics of stochastic mutation ﬂows, etc., need further

demonstration and inspection. All of them require more discussion and study

for the future development of the theory.

11.2 Alignment of Protein Three-Dimensional Structures

How to determine the similarity of the three-dimensional structures of diﬀerent

proteins, whether the method is reasonable and whether the expression of the

similarity is valid are problems to be discussed and solved in the alignment of

protein three-dimensional structures.

11.2.1 Basic Idea and Method of the Alignment

of Three-Dimensional Structures

Alignment of Protein Three-Dimensional Structures

The traditional method of studying the homology of protein three-dimensional

structures uses space movement and rotation [21, 51] to achieve the optimal

ﬁtting of every atom’s point. This method is eﬃcient for comparatively short

local polypeptide chains (such as domain structures), but not so eﬃcient for

longer protein sequences. As movement and rotation are simple rigid transfor-

mations, they are not suitable for the complex comparison of protein three-

dimensional structures. In this book, we try to use the sequence alignment

algorithm to discuss the similarity of protein three-dimensional structures.

Our main idea is outlined as follows:

308 11 Alignment of Primary and Three-Dimensional Structures of Proteins

1. The key to discussing protein three-dimensional structures is the three-

dimensional structure of their backbones. Protein backbones can be con-

sidered to be spliced by a series of triangles, which are called the triangle

splicing belts of protein backbones.

2. In the triangle splicing belt of a protein backbone, the length of each

edge is relatively constant. The key to determining the conﬁguration of

a protein backbone is the dihedral angle between two neighboring trian-

gles, and the value of the dihedral angle ranges within the interval [−π, π)

(or [0, 2π)). Thus the key to determining the conﬁguration of a protein

backbone is the sequence of the dihedral angles between two neighboring

triangles. This sequence is called the dihedral angle sequence of a protein

three-dimensional structure.

3. Homology alignment of protein three-dimensional structures can be car-

ried out using the dihedral angle sequences of the protein three-dimensional

structures, from which the determination of the homology of diﬀerent

protein three-dimensional structures can be obtained. Using these cal-

culations, the homology of protein three-dimensional structures can be

classiﬁed and searched.

We have illustrated above that the key to the determination of protein three-

dimensional structures is the three-dimensional structure of their backbones,

while the three-dimensional structure of the backbones can be spliced by a se-

ries of triangles. In the triangle splicing belt, the key to determining the con-

ﬁguration of the protein backbone is the dihedral angle sequence between two

neighboring triangles, which are denoted as Ψ = {ψ

,ψ

, ··· ,ψ

n−2

},whereψ

is the angle between triangles Δ

and Δ

i+1

, whose values range in the interval

V =[−π, π)(orV =[0, 2π)). We call Ψ the torsion angle sequence of this

protein three-dimensional structure. In this chapter, the alignment of protein

three-dimensional structures represented by their torsion angle sequences. We

denote it as V

= V ∪{−}and call it the expanding set of V .

Penalty Function of the Torsion Angle Sequences

Let Ψ =(ψ

,ψ

, ··· ,ψ

), and Φ =(φ

,φ

, ··· ,φ

) be the torsion angle

sequences of two proteins, where ψ

and φ

are the radian angles whose values

range in (−π, π). We now give the deﬁnition of the penalty functions of the

radian angle and the insertion symbol in ψ



,φ



∈ V

. We divide them into two

types, discrete and consecutive. The deﬁnitions of their penalty functions are

given as follows:

1. Penalty function of the discrete type. If the set V is divided into four

parts [0,π/2), [π/2,π), [−π, −π/2), [−π/2, 0), these would be the four

phases of Cartesian coordinates. If the value of angle ψ is determined by

its phase, its range is given by V

= {0, 1, 2, 3}. The penalty function on

11.2 Alignment of Protein Three-Dimensional Structures 309

set V

= {0, 1, 2, 3, −} will then be

d(ψ



,φ



⎧

⎪

⎨

⎪

⎩

|ψ



− φ



|(mod 2) , if ψ



,φ



∈ V

2 , if one of ψ, φ contains virtual symbol “−”,

but the other does not contain virtual

symbol “−”,

0 , if both of the angles ψ,φ contain virtual

symbol “−”.

(11.6)

2. Penalty function of the consecutive type. If the values of angles ψ



,φ



range within V

, the penalty function will be

d(ψ



,φ



⎧

⎪

⎨

⎪

⎩

|ψ



− φ



|, if neither of the angles ψ, φ contains virtual

symbol “−”,

π, if one of the angles ψ, φ contains virtual

symbol “−”,

and the other does not contain virtual

symbol “−”,

0 , if both of the angles ψ,φ contain virtual

symbol “−”.

(11.7)

3. Denote by Ψ



= {ψ



s,1

,ψ



s,2

, ··· ,ψ



s,n



}, s =1, 2, two sequences whose

lengths are the same and whose values range within either V

or V

.The

total value of the penalty function of Ψ



,Ψ



is then deﬁned as

d (Ψ



,Ψ







i=1





1,i

,ψ



2,i



, (11.8)

where d(ψ



1,i

,ψ



2,i

) is deﬁned in (11.6) or (11.7) according to the discrete

or consecutive type.

The deﬁnitions in (11.6) and (11.7) can be easily rewritten for scoring

functions. For example,

w(ψ



,φ





2 − d(ψ



,φ



) , if ψ



,φ



∈ V

π − d(ψ



,φ



) , if ψ



,φ



∈ V

(11.9)

Minimum Penalty Alignment of the Torsion Angle Sequences

We denote by Ψ

= {ψ

s,1

,ψ

s,2

, ··· ,ψ

s,n

−2

}, s =1, 2 the torsion angle se-

quences of two protein three-dimensional structures, where n

,s=1, 2are

the lengths of these two proteins. Here, Ψ



and Ψ



are two sequences whose

values range within V

or V

.Wecall(Ψ



,Ψ



) the alignment sequences of

310 11 Alignment of Primary and Three-Dimensional Structures of Proteins

(Ψ

,Ψ

) if sequences Ψ



and Ψ



are the same in length and they change into

sequences Ψ

,Ψ

after deleting symbols “−”.

We call (Ψ

∗

,Ψ

∗

) the minimum penalty alignment sequences (or minimum

penalty alignment) of (Ψ

,Ψ

), if (Ψ

∗

,Ψ

∗

) are alignment sequences of (Ψ

,Ψ

)

and d(Ψ

∗

,Ψ

∗

) ≤ d(Ψ



,Ψ



)holdsforany(Ψ



,Ψ



) which are the alignment

sequences of (Ψ

,Ψ

For the penalty functions of ﬁxed torsion angle sequences (Ψ

,Ψ

)and

(11.8), their minimum penalty alignment sequences (Ψ

∗

,Ψ

∗

) can be obtained

by the Smith–Waterman dynamic programming algorithm.

Measurement Index for the Homology of Protein

Three-Dimensional Structures

From the minimum penalty alignment sequences (Ψ

∗

,Ψ

∗

) of torsion angle

sequences (Ψ

,Ψ

), the measurement index for their similarity can be obtained.

It is deﬁned as follows:

1. Denote the total homology measurement index by the total value of the

minimum penalty alignment d(Ψ

∗

,Ψ

∗

) for the torsion angle sequences

(Ψ

,Ψ

2. The total homology measurement index d(Ψ

∗

,Ψ

∗

) can be decomposed as

follows. Denote by k

∗

the total number of inserted symbols in sequences

(Ψ

∗

,Ψ

∗

), and deﬁne

(Ψ

∗

,Ψ

∗

)=d(Ψ

∗

,Ψ

∗

) − k

∗

π (11.10)

to be the measurement index of the local homology of the three-dimen-

sional structures of proteins 1 and 2.

3. It is known from the alignment algorithm that the minimum penalty align-

ment (Ψ

∗

,Ψ

∗

) of torsion angle sequences (Ψ

,Ψ

) may be nonexclusive.

We then take the maximum value k

∗

of the total number of inserted

symbols in the minimum penalty alignment sequences, and d

(Ψ

∗

,Ψ

∗

)in

formula (11.8) would be the maximum value of the local similarity mea-

surement indices.

From this, we ﬁnd that [k

∗

(Ψ

∗

,Ψ

∗

)] are the two measurement indices for

the homology of protein three-dimensional structures. They are the decompo-

sition of the total homology measurement index d(Ψ

∗

,Ψ

∗

11.2.2 Example of Computation in the Discrete Case

Analysis of Protein Three-Dimensional Structures

with Multiple Models

In the PDB database, because of diﬀerent measurements performed for its

structure determination, a protein may have multiple models. For example,

11.2 Alignment of Protein Three-Dimensional Structures 311

Table 11.1. Spatial phase characteristic alignment for protein PDB1BF9

Model Stable region of the torsion Phase Atom site of origin

angles of amino acids and terminus of the

stable region

1 222 020120111213333301133333300 11013223000 33–404

2 131 000320111233333201133220300 13013223020 607–979

3 122 020120111213333201133333300 11011203011 1181–1552

4 010 000120111213333201133333300 13011203011 1755–2126

5 010 020120111213333301133333300 11003203000 2329–2700

6 131 020120111213333201133333300 30101223012 2903–3274

7 113 020130111213333301133333300 31003223010 3477–3848

8 133 020130111213333201133333300 31011223013 4051–4422

9 001 020120111213333201133333300 13011223011 4625–4996

10 220 020120111233332301133332300 13013223020 5199–5570

11 111 000120111213333201133333300 13013223022 5773–6144

12 303 020120111213333301133333300 31011223030 6347–6718

13 303 020120111213333311123220300 13013223031 6921–7292

14 022 000120111213333301133333300 31011223032 7495–7866

15 201 020120111213333201133333300 11101223001 8069–8440

16 021 020130111213333201133333300 11011203021 8643–9014

17 002 020120111213333201133333300 30101223011 9217–9588

18 011 000130111233333201133333300 31011223001 9791–10162

19 000 020120111213333301133333300 11101203010 10365–10736

20 220 000120111213333201133333300 11011203021 10939–11310

21 102 020120111213333301133333300 31013223020 11513–11884

22 103 020130111233332300133333300 31001223002 12087–12458

23 202 020130 11121

{z}

β-sheet

3333301133333300

{z }

Down α-helix

| {z }

Bonding region

30101223 12661–13032

protein PDB1BF9 has 23 models, and each model consists of 41 amino acids.

All of their primary structures are the same, while their spatial coordinates

are diﬀerent (see ﬁle PDB1BF9, PDB database). For the backbones of these

23 models, we compute the value of their phase characteristics, shown in

Table 11.1.

In Table 11.1, it is obvious that the phase characteristic sequences have

bonding regions and β sheet regions. The bonding regions and α-helix regions

of these 23 models can be visualized using Pymol, Rasmol, and other software

on the data ﬁle PDB1BF9. We can see that the three-dimensional structures

of their backbones are similar, especially in the stable regions.

In Table 11.2, the values of torsion angles range in the interval V =[0, 2π).

We can see that at sites 3–30, stable regions are formed.

312 11 Alignment of Primary and Three-Dimensional Structures of Proteins

Table 11.2. Torsion angle data for the 23 models of protein PDB1BF9

1 3.9 4.6 4.9 5.4 4.4 5.8 1.7 3.1 4.9 2.5 5.6 1.1 3.4 1.8 2.4 3.1 2.3 3.2 1.7 5.6

2 6.2 2.7 1.1 5.6 5.0 5.7 2.2 3.1 4.7 2.9 5.4 1.0 3.5 2.0 2.3 2.7 2.5 3.6 1.9 5.5

3 1.7 3.7 3.3 5.5 3.9 5.8 1.8 3.0 4.8 2.9 5.4 0.9 3.6 2.4 2.1 2.5 2.7 3.3 1.9 5.6

4 4.1 1.3 5.5 5.8 4.9 5.6 2.1 3.0 5.0 3.1 5.3 0.8 3.8 2.4 2.3 2.6 2.6 3.3 1.9 5.6

5 1.5 5.0 0.4 5.6 4.6 5.9 1.8 3.0 4.7 2.6 5.6 1.1 3.4 1.9 2.3 2.8 2.5 3.3 1.9 5.6

6 0.7 2.3 0.9 5.5 4.6 5.9 1.9 2.9 4.6 2.7 5.6 1.1 3.5 2.0 2.3 2.7 2.6 3.4 1.8 5.6

7 1.2 1.1 2.4 5.4 4.2 6.2 1.3 2.9 4.9 3.0 5.3 1.0 3.6 2.1 2.2 2.9 2.6 3.3 1.9 5.6

8 1.2 2.0 2.4 5.4 4.1 6.0 1.4 2.9 4.8 2.7 5.5 1.0 3.5 2.1 2.3 2.7 2.7 3.3 2.0 5.5

9 4.9 4.7 1.4 5.6 4.6 5.8 1.8 3.1 4.8 2.9 5.4 1.0 3.5 2.2 2.2 2.7 2.7 3.4 1.9 5.5

10 4.4 3.6 5.0 5.4 4.1 5.9 1.6 3.1 4.6 2.3 5.9 1.0 3.5 1.9 2.2 3.0 2.5 3.1 2.1 5.4

11 5.2 2.5 1.2 5.8 4.7 5.8 2.0 2.8 4.9 2.9 5.4 0.7 3.8 2.3 2.3 2.6 2.6 3.3 2.0 5.6

12 2.2 5.3 2.6 4.7 4.5 5.6 2.0 3.0 5.0 3.1 5.3 0.8 3.7 2.3 2.2 2.7 2.5 3.3 1.8 5.5

13 2.8 4.9 2.7 4.9 4.8 5.4 2.1 3.1 4.9 2.8 5.3 0.8 3.7 2.2 2.4 2.5 2.6 3.5 1.0 0.3

14 0.3 3.4 3.8 5.3 4.7 5.7 2.0 2.8 5.0 3.0 5.3 0.8 3.8 2.4 2.2 2.6 2.7 3.3 1.9 5.6

15 4.0 5.1 1.6 5.7 4.5 5.6 1.8 3.2 4.7 2.7 5.6 1.0 3.5 2.1 2.2 2.8 2.6 3.3 2.0 5.6

16 5.9 3.3 2.1 5.1 4.1 5.9 1.7 2.8 5.1 3.1 5.4 0.8 3.7 2.2 2.3 2.9 2.6 3.4 1.9 5.6

17 5.0 0.5 3.1 5.3 3.9 6.0 1.5 2.9 4.6 2.7 5.6 1.0 3.6 2.3 2.1 2.7 2.5 3.6 1.9 5.5

18 5.0 1.8 0.6 5.5 4.7 6.0 1.8 2.9 5.2 2.7 5.2 1.1 3.2 2.1 2.4 2.8 2.3 3.5 1.9 5.6

19 5.9 4.8 1.0 5.5 4.6 5.6 2.1 3.0 4.9 3.0 5.4 0.8 3.6 2.2 2.4 2.6 2.5 3.2 2.0 5.5

20 4.5 4.4 5.3 5.7 5.0 5.4 2.0 3.2 4.9 2.9 5.3 0.8 3.7 2.4 2.1 2.7 2.5 3.6 1.8 5.5

21 0.4 4.5 4.3 5.3 3.8 5.8 1.6 3.1 4.9 2.9 5.4 1.0 3.6 2.4 2.0 2.7 2.4 3.4 1.8 5.7

22 0.2 5.1 2.6 5.5 3.7 0.1 1.1 2.9 4.6 2.8 5.5 1.0 3.5 2.0 2.2 2.8 2.7 3.4 2.2 5.4

23 4.4 5.3 4.3 5.3 4.3 5.9 1.7 2.9 4.8 2.9 5.4 1.0 3.6 2.2 2.1 2.9 2.5 3.2 1.9 5.5

|4 Stable region |20

1 1.0 1.0 2.6 2.6 2.5 2.9 1.8 1.6 5.2 5.9 2.7 5.1 1.3 5.9 1.9 3.8 5.1 3.0

2 1.2 1.2 2.8 2.6 2.8 3.2 1.1 1.5 5.1 5.8 3.2 4.9 0.9 6.1 2.1 3.9 4.9 2.8

3 0.9 1.4 2.7 2.5 2.6 3.0 1.7 1.4 5.2 5.8 3.0 4.8 1.3 6.1 1.8 3.8 5.2 3.0

4 0.9 1.4 2.7 2.6 2.7 3.0 1.7 1.4 5.2 5.8 3.0 4.9 1.1 6.1 1.7 3.9 5.2 2.8

5 1.0 1.2 2.7 2.5 2.5 2.9 1.8 1.5 5.1 5.8 2.7 5.1 1.4 5.8 1.9 3.8 5.2 3.0

6 0.9 1.4 2.7 2.5 2.5 2.9 1.6 1.5 5.2 5.8 3.2 4.3 1.7 5.8 1.8 4.0 5.1 2.8

7 0.9 1.4 2.6 2.6 2.4 3.0 1.8 1.5 5.2 5.7 2.9 4.7 1.6 5.9 1.8 4.0 4.9 2.9

8 1.0 1.5 2.8 2.4 2.5 3.0 1.6 1.5 5.1 5.7 3.0 4.7 1.6 5.9 1.8 4.0 4.9 2.9

9 1.2 1.3 2.6 2.5 2.6 3.1 1.7 1.5 5.2 5.9 3.0 4.8 1.2 5.9 1.8 3.9 5.1 2.7

10 0.5 1.9 2.6 2.7 3.0 2.8 1.0 1.3 5.1 5.7 3.2 4.9 0.9 6.1 2.0 3.9 4.7 2.8

11 1.0 1.5 2.8 2.6 2.7 3.3 1.6 1.4 5.1 5.7 3.0 5.1 0.9 6.1 2.1 3.8 4.7 2.6

12 1.2 1.0 2.7 2.6 2.6 3.1 1.6 1.4 5.2 5.7 3.0 4.6 1.4 6.2 1.6 4.0 5.1 2.9

13 5.5 2.5 2.6 2.7 2.9 3.2 1.1 1.4 5.1 5.7 3.2 5.0 0.7 6.2 2.0 3.8 4.7 2.6

14 0.9 1.5 2.7 2.6 2.6 3.1 1.7 1.4 5.2 5.7 3.0 4.7 1.4 6.1 1.6 3.9 5.1 3.0

15 1.0 1.4 2.8 2.5 2.4 3.0 1.8 1.6 5.1 5.9 2.7 5.1 1.3 5.8 1.9 3.9 5.1 3.0

16 1.0 1.3 2.8 2.6 2.6 3.0 1.8 1.4 5.2 5.8 2.9 4.7 1.3 6.1 1.6 3.9 5.2 2.9

17 1.2 1.2 2.9 2.4 2.4 3.0 1.7 1.4 5.2 5.7 3.1 4.5 1.8 5.7 1.9 3.9 5.0 2.9

18 1.1 1.0 2.7 2.6 2.6 3.1 1.5 1.5 5.2 5.7 3.0 4.7 1.6 6.0 1.8 3.9 5.1 2.9

19 1.1 1.1 2.8 2.5 2.6 3.1 1.7 1.5 5.2 5.7 3.2 4.7 1.4 5.8 1.8 3.9 5.1 2.8

20 1.1 1.1 2.7 2.6 2.6 3.1 1.7 1.4 5.3 5.7 3.0 4.9 1.1 6.1 1.7 3.9 5.1 2.8

21 0.9 1.2 2.7 2.4 2.4 3.0 1.8 1.5 5.2 5.8 3.0 4.7 1.4 6.0 1.8 4.0 4.9 2.9

22 0.4 2.2 2.8 2.4 2.4 3.0 1.5 1.5 5.2 5.7 3.0 4.5 1.6 6.0 1.7 4.1 5.0 2.9

23 1.1 1.0 2.7 2.4 2.4 3.1 1.7 1.4 5.2 5.7 3.1 4.4 1.7 5.9 1.7 4.1 4.8 2.8

|21 Stable region |30

Remark 13. Table 11.3 is a statistical table for the homology of the alignment

of total spatial torsion angle phase sequences (all the torsion angles of each

model) using the ﬁrst 15 models of protein PDB1BF9. It was computed with

the scoring function in formula (11.9).

Remark 14. Table 11.4 is the statistical table for the homology on the align-

ment of stable regions of the spatial torsion angle phases using the ﬁrst 15

models of protein PDB1BF9. It was computed with the scoring function in

formula (11.9).

11.2 Alignment of Protein Three-Dimensional Structures 313

Table 11.3. Statistical table for the homology of the alignment of total torsion

angle phase sequences of protein PDB1BF9

Model1234567891011121314

20.71

3 0.85 0.68

4 0.76 0.71 0.88

5 0.88 0.66 0.80 0.83

6 0.73 0.68 0.80 0.76 0.73

7 0.83 0.66 0.78 0.73 0.85 0.80

8 0.78 0.68 0.85 0.78 0.73 0.85 0.88

9 0.80 0.73 0.88 0.90 0.78 0.83 0.76 0.83

10 0.85 0.76 0.73 0.73 0.78 0.66 0.73 0.68 0.76

11 0.80 0.83 0.80 0.85 0.78 0.80 0.78 0.78 0.85 0.78

12 0.85 0.66 0.80 0.76 0.80 0.78 0.85 0.85 0.83 0.76 0.76

13 0.73 0.73 0.68 0.68 0.68 0.61 0.68 0.66 0.76 0.71 0.71 0.78

14 0.85 0.66 0.83 0.80 0.78 0.78 0.78 0.80 0.80 0.73 0.80 0.88 0.68

15 0.83 0.66 0.83 0.78 0.80 0.85 0.76 0.78 0.88 0.71 0.78 0.80 0.68 0.76

Table 11.4. Statistical table for the homology of the alignment of stable regions of

the torsion angle phases of protein PDB1BF9

Model1234567891011121314

20.74

3 0.96 0.78

4 0.93 0.81 0.96

5 1.00 0.74 0.96 0.93

6 0.96 0.78 1.00 0.96 0.96

7 0.96 0.70 0.93 0.89 0.96 0.93

8 0.93 0.74 0.96 0.93 0.93 0.96 0.96

9 0.96 0.78 1.00 0.96 0.96 1.00 0.93 0.96

10 0.89 0.74 0.85 0.81 0.89 0.85 0.85 0.81 0.85

11 0.93 0.81 0.96 1.00 0.93 0.96 0.89 0.93 0.96 0.81

12 1.00 0.74 0.96 0.93 1.00 0.96 0.96 0.93 0.96 0.89 0.93

13 0.81 0.78 0.78 0.74 0.81 0.78 0.78 0.74 0.78 0.74 0.74 0.81

14 0.96 0.78 0.93 0.96 0.96 0.93 0.93 0.89 0.93 0.85 0.96 0.96 1.00

15 0.96 0.78 1.00 0.96 0.96 1.00 0.93 0.96 1.00 0.85 0.96 0.96 0.78 0.93

We can see from Table 11.3 that the homologies between Model 1 and Model 2

or Model 5 are 0.71 and 0.88, respectively. It tells us that the homology

between Model 1 and Model 5 is higher than that between Model 1 and

Model 2.