Shen S., Tuszynski J.A. Theory and Mathematical Methods for Bioformatics
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294 10 3D Structure of the Protein Backbone and Side Chains
Fig. 10.4. Tetrahedrons V
O
,V
B
Structural Analysis on Tetrahedrons V
O
,V
B
When the protein backbone L is given, the locations of O
i
are uniquely de-
termined by the parameters {r
2i
,r
4i
,r
6i
,ϑ
i
},whereϑ
i
is the mirror reflection
value of the four-atom points A, C, N
, O, which has been defined in Chap. 8.
Here we have the following formulas:
1. Formula for the volume of tetrahedron V
O
:
V
O
=
1
6
7
7
7
7
7
7
x
1
y
1
z
1
x
2
y
2
z
2
x
3
y
3
z
3
7
7
7
7
7
7
, its absolute value . (10.38)
2. Formula for the area of triangle δ(NAC) has been given in Sects. 10.1 and
10.2, which is denoted as S = S(NAC). Here we introduce
h
O
=3
V
O
S
,h
B
=3
V
B
S
(10.39)
the height of tetrahedrons V
O
and V
B
, respectively, and V
O
and V
B
in
(10.39) the volumes of the tetrahedrons. If we denote by ϑ
O
, ϑ
B
the mirror
10.3 Structure Analysis of Protein Side Chains 295
reflection values of the tetrahedrons V
O
, V
B
,thenwehave
H
O
= ϑ
O
h
O
,H
B
= ϑ
B
h
B
, (10.40)
representing the directional volumes of tetrahedrons V
O
and V
B
, respec-
tively. The formulas for the mirror reflection value ϑ
O
of the four-atom
points N, A, C, O are
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
ϑ
O
=sgn
−→
NO,
−→
AO,
−→
CO
=sgn
⎛
⎜
⎝
7
7
7
7
7
7
7
x
1
y
1
z
1
x
2
y
2
z
2
x
3
y
3
z
3
7
7
7
7
7
7
7
⎞
⎟
⎠
,
ϑ
B
=sgn
−→
NB,
−→
AB,
−→
CB
=sgn
⎛
⎜
⎝
7
7
7
7
7
7
7
x
1
y
1
z
1
x
2
y
2
z
2
x
3
y
3
z
3
7
7
7
7
7
7
7
⎞
⎟
⎠
,
(10.41)
where x
1
, ··· ,z
3
and x
1
, ··· ,z
3
are defined in (10.36), while
(r
1
, r
2
, r
3
)=r
1
× r
2
, r
3
=
7
7
7
7
7
7
x
1
y
1
z
1
x
2
y
2
z
2
x
3
y
3
z
3
7
7
7
7
7
7
is the mixed product of the three vectors r
1
, r
2
, r
3
.
10.3.2 Statistical Analysis of the Structures
of Tetrahedrons V
O
,V
B
The statistical analysis of the structures of the tetrahedrons V
O
,V
B
refers to
the statistical analysis of the structural parameters of atoms O, B and N, A,
C, N
. Using the PDB-Select database, we may perform statistical analysis on
these structural parameters (Table 10.4).
Structure Analysis of the Type L and Type D Tetrahedron V
B
1. Among the 696,637 amino acids analyzed, the numbers of tetrahedrons
of mirror value +1 or −1are37,682 and 658,955 respectively, thus the
proportion of those with mirror reflection value +1 is 5.41%.
2. In the PDB-Select database, for tetrahedron V
B
,thenumbersoftypeL
and type D tetrahedrons are 653,969 and 141, respectively, hence the
proportion is about 10,000:2. Thus, type D tetrahedrons are rare.
296 10 3D Structure of the Protein Backbone and Side Chains
Table 10.4. Statistics of the basic parameters of tetrahedrons V
O
and V
B
12 3 4567 8 910
R(NA) R(AC) R(CO) R(ON
)R(N
A
) R(NC) R(AO) R(CN
)R(OA
)R(NAC)
Mean 1.46220 1.52459 1.22781 2.24328 1.46211 2.45620 2.40675 1.33654 2.35253 2.07139
Variance 0.00019 0.00016 0.00021 0.00147 0.00017 0.00260 0.00058 0.00237 0.16705 0.00296
Standard deviation 0.01371 0.01251 0.01454 0.03830 0.01300 0.05103 0.02400 0.04871 0.40872 0.05440
11 12 13 14 15 16
R(ACO) R(ACN
)R(CN
A
) ψ
1
ψ
2
ψ
3
Mean 1.61603 1.81455 1.65359 0.38551 0.47297 3.15149
Variance 0.00094 0.00294 0.00321 3.75848 2.73928 0.03761
Standard deviation 0.03066 0.05421 0.05666 1.93868 1.65508 0.19392
12 3 4567 8 910
R(NB) R(AB) R(CB) R(NB) R(NAC) R(ACB) V(NACB) S(NAC) H(NACB)
Mean 2.44942 1.53138 1.43440 2.45479 2.07516 1.91471 −1.41089 −0.42143 1.03758 1.21819
Variance 0.00178 0.00060 0.19462 0.00257 0.00270 0.22472 0.88474 0.00059 0.00068 0.00240
Standard deviation 0.04217 0.02460 0.44115 0.05073 0.05201 0.47404 0.94061 0.02430 0.02600 0.04894
11 12 13 14 15 16
R(ACO) R(ACN
)R(CN
A
) ψ
1
ψ
2
ψ
3
Mean 1.61603 1.81455 1.65359 0.38551 0.47297 3.15149
Variance 0.00094 0.00294 0.00321 3.75848 2.73928 0.03761
Standard deviation 0.03066 0.05421 0.05666 1.93868 1.65508 0.19392
V
O
is the volume of the tetrahedron V (NACO), S(NAC) is the area of the bottom triangle δ(NAC),
h
O
is the height of the tetrahedron V (NACO) with regard to the bottom triangle δ(NAC)
10.4 Exercises, Analyses, and Computation 297
10.4 Exercises, Analyses, and Computation
Exercise 48. Attempt the following calculation on the atoms of all the pro-
tein backbones in the PDB-Select database:
1. For every atomic sequence in the protein backbone L
1
(see the defini-
tion in formula (10.18)), calculate the parameter sequences. The relevant
parameters are defined in formula (10.21).
2. For every atomic sequence in the protein backbone L
2
(see the defini-
tion in formula (10.18)), calculate the parameter sequences. The relevant
parameters are defined in formula (10.21).
3. For every atomic sequence in the protein backbone L
2
(see the definition
in formula (10.18)), where A
i
are replaced by N
i
or C
i
separately, calculate
the parameter sequences. The relevant parameters are defined in formula
(10.21).
Exercise 49. Based on the calculation in Exercise 48, provide the following
discussion and analysis:
1. Analyze the stability of each parameter, and calculate its mean, variance
and standard deviation.
2. In the protein backbone L
2
, when the atomic sequences are A
i
,N
i
,and
C
i
, respectively, compare the correlation of each parameter sequence.
3. It can been seen from the above analysis that the torsion angle se-
quences ψ
i
and mirror sequences ϑ
i
are unstable. Attempt a histogram
analysis of the torsion angle sequences ψ
i
and attempt to perform fre-
quency analysis on the mirror sequences ϑ
i
.
Exercise 50. From the calculation in Exercise 48, task 3, we obtain the tor-
sion angle sequence ψ
i
and mirror sequence ϑ
i
for the atomic sequence A
i
,and
therefore obtain the phase sequence ϑ
i
(see the definition in formula (10.26)).
Perform the following calculations:
1. Compare the statistical properties of the phase sequence ϑ
i
with the sec-
ondary structure sequence (see the definition and explanation in formula
(9.1) and Table 9.1). For instance, the joint distribution of (ϑ
i
,η
i
), prop-
erties such as the interprediction between ϑ
i
and η
i
.
2. For the phase sequence ϑ
i
, there is also the problem of predicting ϑ
i
from
the primary structure sequence ξ
i
. Attempt to calculate the informational
statistics table of the primary structure ξ
i
and the phase sequences ϑ
i
using the method in Chap. 9, and predict the phase sequences from the
primary structure sequences.
Exercise 51. Attempt the following calculations on the atoms of all the pro-
tein side chains in the PDB-Select database:
1. For each amino acid in the database, calculate each parameter for the
tetrahedron structure V
B
= {N, A, C, B}.
298 10 3D Structure of the Protein Backbone and Side Chains
2. For the pairs of conjoined amino acids in the database, calculate each
parameter for the five-atom structure {N, A, C, O, N
},whereN
is the
nitrogen atom of the next amino acid.
3. For the pairs of conjoined amino acids in the database, calculate each
parameter for the six-atom structure {N, A, C, B, O, N
},whereN’isthe
nitrogen atom of the next amino acid.
4. For the pairs of conjoined amino acids in the database, calculate each
parameter for the eight-atom structure {N, A, C, B, N
, A
, C
, B
},where
N
,A
,C
,B
are the corresponding atoms of the next amino acid.
5. For different amino acids and dipeptide chains, calculate the structural
data for the atomic distances.
Exercise 52. Based on the calculation in Exercise 51, provide the following
discussion and analysis:
1. Analyze the stability of each parameter, and calculate its mean, variance,
and standard deviation. Also, discuss the similarities and differences be-
tween distinct amino acids.
2. From this, determine the structural characteristics of the related atoms,
e.g., four points on the same plane, the unicity of the tetrahedron
phase V
B
.
Exercise 53. Based on the calculation in Exercises 51 and 52, provide the fol-
lowing discussion and analysis for different amino acids and dipeptide chains:
1. For different dipeptide chains, based on the calculation in Exercise 51,
task 4, construct the second-rank topological structure map for eight-atom
points {N, A, C, B, N
, A
, C
, B
}.
2. For different amino acids and dipeptide chains, based on the calculation in
Exercise 51, task 4, construct the second-rank topological structure map
for all the atom points.
3. Together with the classification in Table 10.4, discuss the similarities and
differences between the second-ranked topological structure map.
11
Alignment of Primary and Three-Dimensional
Structures of Proteins
In the previous chapter, protein three-dimensional structures and a method
for predicting protein secondary structures using informatics and statistics
have been introduced. In this chapter, we focus on the issues of alignment
algorithms for the three-dimensional structure of proteins.
11.1 Structure Analysis for Protein Sequences
In biology, sequence analysis refers to analyzing DNA (or RNA) sequences and
protein sequences. In the above, we mainly focused our discussion on DNA
(or RNA). Here we introduce structure analysis for protein sequences.
11.1.1 Alignment of Protein Sequences
Structure analysis for proteins is a huge area of active research. It can be
divided into the following aspects.
Research on Protein Primary Structures
It is known that proteins are formed by binding different amino acid residues,
and protein primary structures are the sequences consisting of different per-
mutations of these amino acids. Thus, the study of protein primary structures
concerns the sequence structures of these different amino acids’ permutations.
The issues are:
1. Structure analysis and alignment of protein sequences. Problems such as
mutation and reorganization, etc. are relevant to our understanding of pro-
tein structures, and the evolution and classification of protein structures
are also related. Thus, sequence alignment is still a basic problem.
300 11 Alignment of Primary and Three-Dimensional Structures of Proteins
2. Grammar and word-building analysis for protein sequence structures. If
protein sequences are disassembled, many small protein sequence segments
result. These segments are called characteristic chains in biology. If peptide
chains are considered to be the basic building blocks for protein structure
and function, they would correspond to words and phrases of a book. In
order to get an overview of the structure and functions of proteins, one
approach is to study the grammar and syntax. That is, to set up special
databases (databases of grammar and syntax) of the peptide chains whose
structure and function is known, and then search and compare the general
structures of proteins in these databases, so that the general structures
and functions of these proteins can be determined. This problem is still
related to sequence alignment, where the segments are usually short in
length and large in number.
Research on Protein Three-Dimensional Structures
and Their Functions
Protein three-dimensional structure includes secondary structure and other
higher level structures. From biological research, we know that the activity and
function of a protein is related to its three-dimensional structure. If a protein
is heated, it will be denatured and thus lose its activity. It is found biologically
that, for the proteins whose primary structures are different, if they have the
same three-dimensional structures, their functions may be similar. Therefore,
the research on protein three-dimensional structures is closely related to that
of their functions.
The topic of protein three-dimensional structures covers many aspects.
They are:
1. Research on protein secondary structures. In biology, peptide chains with
special structural characteristics are called secondary structures, such as
α helices and β sheets. These structures are common units of protein
general three-dimensional structures, and their combinations determine
the functions of the proteins they make up. Therefore, the research on
protein secondary structures is essential to that of the three-dimensional
structures. The analysis of protein secondary structures focuses on their
prediction, which is to confirm protein secondary structure characteristics
according to their primary sequences.
Many mathematical tools and methods have been used in the research on
protein secondary structure prediction, while so far, the accuracy of the
predictions is far from satisfactory. In the past ten years, the accuracy
of the predictions of α helices and β sheets remained at around 70%. For
some fixed proteins whose types are known, the accuracy of the prediction
has improved. However, because the types are fixed, their definition of the
prediction accuracy is not as good.
2. The main unresolved difficulty present in the analysis of protein three-
dimensional structures is the long distance folding of the three-dimensional
11.1 Structure Analysis for Protein Sequences 301
structures. That is, under what condition will the protein three-dimen-
sional structures begin to fold back, and to which part of the protein
will they join. Although there is much research involved in studying this
problem, no efficient method has yet been found.
3. Research on the relationship between protein three-dimensional structures
and their functions. This problem has always been both of great interest
and fraught with difficulties in bioinformatics. Since the formulation of the
proposal of the proteome project, the research on the relationship between
protein three-dimensional structures and their functions has focused on
the interactions between different proteins and the relationship between
these interactions and protein three-dimensional structures.
Generally speaking, the research on protein three-dimensional structures and
the relationship between protein three-dimensional structures and their func-
tions is still in an early stage of development. So far, large amounts of data
have been accumulated and relevant information for medicine and biology is
being drawn upon. However, there is still a long way to go and many concepts
need to be developed until mature applications are carried out. These are the
areas where mathematics will play an important role, although we will not
discuss them in detail in this book.
11.1.2 Differences and Similarities Between the Alignment
of Protein Sequences and of DNA Sequences
Above, we specified some of the problems in protein structure research. We
know from existing methods that although some are just in the early stages,
many are related to sequence alignment, and especially to multiple sequence
alignment which plays an important role. However, some differences lie be-
tween the alignment of protein sequences and that of DNA sequences; thus
the discussion in the above chapters does not directly translate to the com-
putation of the alignment of protein sequences. The differences between the
alignment of protein sequences and that of DNA sequences are described next.
Differences Between the Constitution of Protein Sequences
and that of DNA Sequences
1. Proteins usually consist of the 20 commonly occurring amino acids, thus
they are more complex than DNA which consists of only four kinds of nu-
cleotides. In the alignment of DNA sequences, we assume the distributions
of sequences follow a uniform distribution. This assumption is suitable for
DNA sequences, but not for protein sequences. According to our statistical
analysis and computation, even in the Swiss-Prot database, the statistics
of single amino acids does not follow a uniform distribution. For the statis-
tics on polypeptide chains, the differences between their distributions are
even greater.
302 11 Alignment of Primary and Three-Dimensional Structures of Proteins
2. The length of a protein sequence is usually not large. For some long protein
sequences, the lengths are in the thousands of residues. Thus in the pair-
wise alignment of sequences, an increase in the computational complexity
will not be the main issue. Therefore, the advantages of using the SPA
algorithm over the dynamic programming algorithm will not be immedi-
ately obvious. However, when it comes to multiple sequence alignment,
the design of fast algorithms will still be a key issue.
3. The penalty functions for the alignment of DNA sequences are simpler,
either in the Hamming matrix or the WT-matrix as they both involve the
strong symmetry condition. The penalty functions used in the alignment
of protein sequences are the PAM matrix series and BLOSUM matrix
series, neither of which involves the strong symmetry condition, or even
the symmetry condition. Therefore, it must be further examined whether
the series of limit theorems given in this book will hold for the alignment
of protein sequences.
Similarities Between the Alignment of Protein Sequences
and that of DNA Sequences
The main similarity between the alignments of protein sequences and of DNA
sequences is that the theory of their modulus structure analysis still applies.
Therefore, the definition and algorithm of uniform alignment remains effective.
These will be discussed in the following text.
11.1.3 The Penalty Functions for the Alignment of Protein
Sequences
The penalty functions used in the alignment of protein sequences are the
PAM and BLOSUM matrix series. We describe their origin and properties as
follows.
Origin of the PAM and BLOSUM Matrix Series
In the alignment of DNA sequences, the Hamming matrix and the WT-matrix
used as the penalty functions are relatively simple, and the settings are easy
to understand. They both have the same characteristic of strong symmetry.
In the Hamming matrix, the errors of the nucleotides occurring in the align-
ment are given the same penalty, while in the WT-matrix, a, g and c, u are
differentiated by giving smaller penalties due to the errors. This is related to
the double helix structure of DNA and the central dogma.
Because of the differences in protein sequences as compared to DNA se-
quences, the determination of their penalty function is more complex. In the
PAM matrix series and the BLOSUM matrix series, it can usually be de-
termined by statistical computation methods. Their values are the negative
11.1 Structure Analysis for Protein Sequences 303
values of the penalty functions; the closer the amino acids are, the higher the
score will be. Thus, we call them scoring matrices. The related steps of the
computation are outlined below:
1. Choose the known homologous sequences to be statistical samples whose
homologies have a specific biological definition. The PAM matrix series
and BLOSUM matrix series are the results of the statistics applied to
different types and scales.
2. Alignment with the selected homologous sequence samples and the result
is constructed for a group of pairwise sequences.
3. Based on the statistics of the selected pairwise sequences, we compute
their joint frequency distribution p(a, b),a,b∈ V
q
,whereq is equal to 20,
23 or else. The symbol p(a, b) denotes the frequency of a, b occurring in
the same position of the pairwise sequences after alignment.
4. According to the frequency distribution p(a, b), a, b ∈ V
q
,wecalculate
their mutual entropy density function:
k(a, b)=log
p(a, b)
p(a)q(b)
,a,b∈ V
q
, (11.1)
where
p(a)=
b∈V
q
p(a, b) ,q(b)=
a∈V
q
p(a, b) .
5. Properly magnify the mutual entropy density function k(a, b), and make
it an integer, to obtain the scoring matrix.
Properties of the Relative Entropy Density Functions
In informatics, it is known that the average of the relative entropy density
function is the Kullback–Leibler divergence.
D
0
=
a,b∈V
q
p(a, b)k(a, b)=
a,b∈V
q
p(a, b)log
p(a, b)
p(a)q(b)
, (11.2)
where D
0
≥ 0 definitely holds, and the equality holds if and only if
p(a, b)=p(a)q(b)
holds for any a, b ∈ V
q
.
Besides the D
0
index, another function is frequently used in bioinformatics,
namely,
D
1
=
a,b∈V
q
p(a)q(b)k(a, b)=
a,b∈V
q
p(a)q(b)log
p(a, b)
p(a)q(b)
. (11.3)