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

Подождите немного. Документ загружается.

334 12 Depth Analysis of Protein Spatial Structure

and deﬁne

=min{π/2 − ψ



,j= k} (12.8)

and j

= k, such that

− ψ



= θ

. We then have the following:

(a) θ

> 0, otherwise a

∈ Π

; this contradicts the condition that there

is only one point a in Π

(b) Point a

must be in A

. Otherwise, we assume that point a

in A

, then rotate Π

around point a with angle θ

= π/2 − ψ



such that vector a

falls in the rotated new plane Π



.Wedenotethe

normal vector of plane Π



by b



, and denote the angle between sphere b

and sphere b



by θ

. For any j = k,j

,wehaveθ

≤ π/2 − ψ



.Now,

following from Lemma 6 and a

∈ Π



,wegetN



= s

(a),

which contradicts the deﬁnition of s

(a). It follows that point a

must

belong to A

is in A

, then similarly to the discussion above, we rotate Π

around point a with angle θ

= π/2 −ψ



, such that vector a

falls in

the rotated new plane Π



By Lemma 6, we get

(π



,a) ≤ n

(π

,a)=s

(a) .

π is then also the depth cut plane of a, which comes back to case 2 and case 3.

Therefore



(a) ≤ n

(Π



,a) ≤ n

(Π

,a)=s

(a)

and s



(a)=s

(a) is true. The theorem follows.

12.1.4 Proof of Theorem 41

We prove this theorem in four steps:

1. Step 12.1.1.IfS

= {a} is a set of a single point, then Theorem 41 is

obviously true.

2. Step 12.1.2.IfS

= {a, b},a = b is a set of dual points, then for an

arbitrary point c on the line segment ab,wehaves

(c)=s

. We will

prove this proposition by contradiction.

Suppose s

(c)=s

is not true; then by the deﬁnition of s

,weget

a, b = c,ands

. By the deﬁnition of depth, we realize that there

must be a plane Π

that is through point c, such that n

(Π

)=s

(c).

Then two cases as follows may happen:

(a) Line segment ab is in plane Π

. By the deﬁnition of depth, we get

(a),a

(b) ≤ n

(Π

) <s

, which contradicts the deﬁnition of points

a, b, so this case will not happen.

(b) Line segment ab is not in plane Π

,thenpointsa, b lie on both sides of

plane Π

. Without loss of generality, we let point a lie outside of plane

. By the property of depth, we realize that s

(a) ≤ n

(Π

) <s

12.2 Statistical Depth Analysis of Protein Spatial Particles 335

true, which contradicts the deﬁnition of point a, so this case will not

happen either.

This shows that s

is not true, so for any c ∈ ab, s

(c)=s

true.

3. Step 12.1.3. Similar to the proof in Step 12.1.2, we can prove that S

must be a convex set. Thus, for any a, b ∈ S

, and any point c on line

segment ab, s

(c)=s

must be true. So S

must be a convex set.

4. Step 12.1.4. By the properties in Sect. 12.1.2, it is impossible to ﬁnd any

noncoplanar four points of set A in S

. We do not give a detailed proof

here. Thus, the theorem is proved.

12.2 Statistical Depth Analysis of Protein Spatial

Particles

Statistical depth analysis of protein spatial particle requires the application

of statistical analysis for depth indicators of spatial structure on all proteins

in the PDB database. There are many types of depth indicators of protein

spatial structure. In this book we choose several speciﬁc depth indicators for

the statistical analysis.

12.2.1 Calculation for Depth Tendency Factor of Amino Acid

Depth Database of Protein

Using the method in Theorem 40 for the PDB database, we can build a depth

database of proteins. Several important symbols are deﬁned as follows:

Let

A = { A

, ··· ,A

} (12.9)

be all the protein sequences in the PDB database. For the sake of simplicity,

we only discuss the proteins whose lengths (number of amino acids) are larger

than 50. Here m is the number of proteins. A

is the ith protein in the PDB

database. We denote the primary structure sequence of protein A

=(b

, ··· ,b

) ,b

∈ Z

in which Z

= {1, 2, ··· , 20} represents the 20 common amino acids. We also

denote the spatial structure sequence of protein A

=(a

, ··· ,a

) ,

in which a

=(x

) represents the coordinate of C

of the jth amino

acid in protein A

The data of sequence b

and a

, i =1, 2, ··· ,m, can both be obtained

from the PDB database. Using these data, we can calculate the depth of each

336 12 Depth Analysis of Protein Spatial Structure

point in the sequence, and we denote s

=(s

, ··· ,s

), where s

) is the depth of jth aminio acid (jth C

atom) of the ith protein. We

refer to

S = {(b

, a

, s

),i=1, 2, ··· ,m} (12.10)

as the depth database of protein. This database is available on our Web

site [99].

Deﬁnition of Depth Tendency Factor of Amino Acids

Through analysis of the depth database of protein, we realize that in protein

spatial structures, distributions of the depths of diﬀerent amino acids are

obviously diﬀerent. We call this phenomenon the depth tendency factor

of the amino acid. There are various ways of deﬁning the tendency factor.

They will be discussed below:

1. Zero depth tendency factor of amino acids. The zero depth tendency

factor of an amino acid is the proportion of each amino acid which appears

on the surface of the protein (zero depth). We distinguish this proportion

into two classes: absolute zero depth tendency factor and relative zero

depth tendency factor:

(a) Absolute zero depth tendency factor: This is the proportion of a cer-

tain amino acid in all the zero depth amino acids.

(b) Relative zero depth tendency factor: For a certain amino acid, this is

its proportion on the protein surface (zero depth).

2. Deepest depth tendency factor of protein. For each protein, there is

one amino acid (sometimes several amino acids) whose depth is the largest.

We call this (these) amino acid(s) the deepest point of the protein.

The deepest depth tendency factor is deﬁned as follows: the proportion

of diﬀerent amino acids in all of these amino acids that have the deepest

depth.

3. Average depth of amino acids. This means the average depth of all

the amino acids in the database. We refer to it as the average depth ten-

dency factor of amino acid. We perform statistical calculations on protein

chains (whose lengths are greater than or at least equal to 50) in the PDB

database, and obtain their tendency factors as given in Table 12.1.

Remark 20. The symbols in Table 12.1 are explained as follows:

1. i is the sequence number of the amino acid, a(i) is the total number of

ith amino acids with zero depth, n



i=1

a(i) is the total number of

amino acids in zero depth, n(i) is the total number of ith amino acids,

n =



i=1

n(i) is the total number of amino acids, m =



i=1

c(i)isthe

total number of the deepest amino acids.

2. Thus, we have the following statistics:

(a) 100n(i)/n: percentage of frequency distribution of the amino acids

12.2 Statistical Depth Analysis of Protein Spatial Particles 337

Table 12.1. Depth tendency factor of 20 common amino acids

One- 0-depth Amino acid DPF Amino acid 0-depth 0-depth Grade DPF Grade

letter F F F F RF ofDP ofDP

code a(i) n(i) c(i) 100n(i)/n 100a(i)/n

100a(i)/n(i) Rel. freq. 100c(i)/m

A 121175 718678 2881 8.202 7.477 16.861 i 8.127 i

R 79441 428179 1015 4.886 4.902 18.553 i 2.863 b

N 105583 390577 1426 4.457 6.515 27.033 h 4.023 b

D 157163 504074 1111 5.753 9.698 31.179 h 3.134 b

C 5766 128879 1108 1.471 0.356 4.474 b 3.126 b

Q 79403 327994 736 3.743 4.900 24.209 h 2.076 b

E 180862 553759 749 6.320 11.160 32.661 h 2.113 b

G 183580 675173 2693 7.705 11.328 27.190 h 7.597 i

H 30892 202744 971 2.314 1.906 15.237 i 2.739 b

I 25836 491050 4139 5.604 1.594 5.261 b 11.676 h

L 45885 763008 4298 8.708 2.831 6.014 b 12.125 h

K 151112 515808 644 5.887 9.325 29.296 h 1.817 b

M 17069 185310 697 2.115 1.053 9.211 b 1.966 b

F 18629 347024 2530 3.960 1.150 5.368 b 7.137 i

P 105018 405990 731 4.633 6.480 25.867 h 2.062 b

S 136614 527753 1511 6.023 8.430 25.886 h 4.263 b

T 99560 512565 2003 5.850 6.144 19.424 i 5.651 i

W 7397 130864 993 1.493 0.456 5.652 b 2.801 b

Y 21818 312638 1074 3.568 1.346 6.979 b 3.030 b

V 43717 619250 4047 7.067 2.698 7.060 b 11.417 h

0-depth F = zero depth frequency, aminoacidF = amino acid frequency, DPF = deepest point frequency,

0-depth RF = relative zero depth frequency, grade of DP = grade of the deepest point

338 12 Depth Analysis of Protein Spatial Structure

(b) 100a(i)/n

: in all amino acids with zero depth, the percentage of ith

amino acid (absolute zero depth tendency factor)

zero depth tendency factor)

(d) 100c(i)/m: in all the deepest points of amino acids, the percentage of

the ith amino acid (absolute deepest tendency factor)

Properties of Depth Tendency Factor

Based on Table 12.1, we can see that the depth tendency factor of an amino

acid has the following characteristics:

1. In the view of the frequency and relative frequency of zero depth, the

proportions of diﬀerent amino acids on the surface of a protein are obvi-

ously diﬀerent. The highest proportion is for the glutamic acid (E), whose

proportion is as high as 32.66%, and the lowest proportion is for cysteine

(C), whose proportion is only 4.47%.

2. Amino acids can also be classiﬁed by zero depth relative frequency. For

example, the amino acid whose relative frequency is more than 20% (or

10–20%, or less than 10%) is said to be of the high (h) (or middle (i), or

low (b)) surface tendency class, as shown in the table.

3. We can also use the depth frequency of the deepest point to classify amino

acids. The amino acid whose depth frequency is above 10% (or 5–10%, or

below 5%), is said to be of the high (h) (or middle (i), or low (b)) deepest

point tendency class, as shown in the table.

12.2.2 Analysis of Depth Tendency Factor of Amino Acid

Relationship Between Depth Tendency Factors

and Other Chemical and Physical Indices

We found that the depth tendency factors are comprehensively related to

chemical and physical indices such as charge, polarity, chemical compound

class, hydrophobicity, etc. (shown in Table 12.1). Correlative data are shown

in Table 12.2.

The second row shown in Table 12.2 is the frequency of amino acids in

the PDB database. The data of hydrophobicity comes from [26, 53, 61, 107].

The symbol +1 indicates electropositivity, −1 indicates electronegativity, and

0 indicates neutrality. In the column “Polarity” for dielectric polarity, +1 and

−1 indicate polar and nonpolar properties, respectively.

In the column “Chemical compound class”, (−1, −1), (1, −1), (1, 1) indi-

cate the acyclic hydroxyl, alkaline, and acidic categories, respectively, (0, 0)

indicates the hydroxy and sulfur-containing category, and (−1, 1) indicates

the aromatic, and acclaimed category.

Using the data in Table 12.2, we can do data ﬁtting on both the depth

factor and the tendency factor. Let Y be the relative zero depth tendency

12.2 Statistical Depth Analysis of Protein Spatial Particles 339

Table 12.2. Chemical and physical indices of amino acids and depth tendency factors

Code Freq. (%) Charge Polarity Chemical K-D Eisenberg Meek F-P Wolfenden Absolute Relative Deepest

in PBD compound 0-depth 0-depth depth

class TF TF TF

A 8.202 0 −1(−1, −1) 1.80(7) 0.25(7) 0.20 0.31 1.49 7.477 16.861 8.127

R 4.886 1 −1(1, −1) −4.0(20) −1.8(20) −1.3 −1.0 −19.9 4.920 18.553 2.863

N 4.457 0 1 (1, 1) −3.5(15) −0.64(16) −0.69 −0.60 −0.97 6.515 27.033 4.023

D −5.753 1 −1(1, 1) −3.5(15) −0.72(18) −0.72 −0.77 −10.9 9.698 31.179 3.134

C 1.471 0 1 (0, 0) 2.5(5) 0.04(9) 0.67 0.54 1.24 0.356 4.474 3.126

Q 3.743 0 1 (1, 1) −3.5(13) −0.69(17) −0.74 −0.22 −9.4 4.900 24.209 2.076

E −6.320 1 −1(1, 1) −3.5(15) −0.62(15) −1.1 −0.64 −10.2 11.160 32.661 2.113

G 7.705 0 1 (− 1, −1) 0.4(8) 0.16(8) 0.06 0.0 2.4 11.328 27.190 7.595

H 2.314 1 −1(1,

−1) −3.2(14) −0.40(14) −0.0 −0.13 −10.3 1.906 15.237 2.739

I 5.604 0 −1(−1, −1) 4.5(1) 0.73(1) 0.74 1.8 2.15 1.594 5.261 11.678

L 8.708 0 −1(−1, −1) 3.8(3) 0.53(4) 0.65 0.17 2.28 2.831 6.014 12.125

K 5.887 1 −1(1, −1) −3.9(19) −1.1(19) −2.0 −0.99 −9.5 9.325 29.296 1.817

M 2.115 0 −1(−1, 1) 1.9(6) 0.26(6) 0.71 1.23 1.48 1.053 9.211 1.966

F 3.960 0 −1(0, 0) 2.8(4) 0.61(2) 0.67 1.79 −0.76 1.150 5.368 7.137

P 4.633 0 −1(0, 0) −1.6(11) −0.07(11) −0.44 −0.72 NM 6.480 25.867 2.062

S 6.023 0 1 (−1, 1) −0.8(10) −0.26(13) −0.34 −0.04 −5.06 8.430 25.886 4.263

T 5.850 0 1 (−1, 1) −0.7(9) −0.18(12) −0.26 −0.34 −4.88 6.144 19.424 5.651

W 1.493 0 −1(0, 0) −0.9(11) 0.37(5) 0.45 2.25 − 5.88 0.456 5.652 2.801

Y 3.568 0 1 (0, 0) −1.3(12) −0.02(10) −0.22 −0.96 −6.11 1.346 6.979 3.030

V 7.067 0 −1(−1, −1) 4.2(2) 0.54(3) 0.61 1.22 1.99 2.698 7.060 11.417

TF = tendency factor, NM =nomention

340 12 Depth Analysis of Protein Spatial Structure

Table 12.3. Table of data ﬁtting of relative zero depth tendency factor of amino

acid

Code Act. val. Reg. est. Fit. err.

A 16.861 18.7888 −1.92777

R 18.553 19.0889 −0.53587

N 27.033 29.1475 −2.11446

D 31.179 30.6515 0.527526

C 4.474 6.92408 −2.45008

Q 24.209 22.7095 1.49953

E 32.661 32.2877 0.373308

G 27.190 25.1997 1.99029

H 15.237 10.1557 5.08133

I 5.261 5.41785 −0.156846

L 6.014 5.56178 0.452215

K 29.296 29.9323 −0.636279

M 9.211 12.3288 −3.1178

F 5.368 0.85603 4.51197

P 25.867 23.9452 1.92178

S 25.886 22.831 3.05504

T 19.424 20.0725 −0.648528

W 5.652 15.427 −9.775

Y 6.979 13.2924 −6.31337

V 7.060 7.57195 −0.511953

act. val. = actual value, reg. est. = regression

estimate, ﬁt. err. = ﬁtting error

factor (in the 12th column), x

be the electric property and polarity (in

the third and fourth columns), (x

) be the data in the ﬁfth column, and

let x

, ··· ,x

be hydrophobicity (in the sixth to tenth columns).

Thus, we can get their linear regression relationship through the formula:

Y =17.1434 − 9.68877x

− 0.714291x

− 5.64749x

− 3.19165x

− 3.25787x

− 13.4022x

− 4.85877x

+1.92445x

+1.05084x

. (12.11)

This ﬁtted result is given in Table 12.3.

Furthermore, we obtain the mean square error, standard deviation, and

mean error of ﬁtting as follows:

⎧

⎪

⎨

⎪

⎩

mean square error σ



i=1

− ˆy

)

=6.29797 ,

standard deviation σ =

√

=2.50958 ,

mean error e =



i=1

− ˆy

| =1.74419 .

(12.12)

The errors in (12.12) are deﬁned by percentage, where y

is the actual zero

depth tendency factor, ˆy

is the result which is ﬁtted by chemical and physical

12.2 Statistical Depth Analysis of Protein Spatial Particles 341

indices of amino acid, and p

is the frequency distribution of the amino acid

in the PDB database.

Following from these data ﬁtting results, we can see that the ﬁtting error

is not too large. Moreover, from ﬁtting formula (12.12), we realize that the

chemical and physical indices of amino acids have diﬀerent eﬀects on the

relative zero depth tendency factor. For example, x

(charge of amino

acids and the hydrophobic factor obtained from F-P and Eisenberg) have

large eﬀects, while x

(polarity of amino acids and the hydrophobic

factor obtained from Wolfenden) have smaller eﬀects. This diﬀers from the

conclusions in some publications which suggest that polarity is the major

factor inﬂuencing the depth of amino acids in proteins.

Statistical Analysis of Both Length and Depth Indicators

of Proteins

From the protein-depth database, we can perform statistical analysis of both

length and depth indicators of proteins. The so-called depth indicators of

proteins include:

1. The number of amino acids on the surface of a protein. For each protein,

it is the total number of amino acids in zero depth.

2. The deepest depth of a protein, this means the depth of the deepest point

in a protein.

3. Average depth of a protein, this means the average depth of all amino

acids in a certain protein.

These three indicators relate to the length of the protein. We denote the length

of the ith protein by X(i)=n

, and denote three depth indicators of the ith

protein by Y

(i), τ =0, 1, 2. In the PDB database, these indicators of proteins

with diﬀerent length are listed in [92]. Using the observation data Y

(i)and

X(i), we can perform regression analysis. Using the least-squares estimation,

we get the linear regression equation of these data:

(i)=α

X(i)+β

,τ=0, 1, 2 , (12.13)

in which the values of the regression coeﬃcients α

and β

are deﬁned by

τ 01 2

0.05628 0.39897 0.06580

31.80478 −11.82391 −2.30257

(12.14)

The regression errors of (12.14) are deﬁned as follows:

τ 012

mean square error





45.2454 97.0313 1.3734

standard error (σ

)6.7265 9.8504 1.1719

absolute error (δ

)5.1237 7.2439 0.8046

(12.15)

342 12 Depth Analysis of Protein Spatial Structure

Fig. 12.6. The ﬁtting relationship between protein length and depth map

where the deﬁnitions of mean square error, standard error, and absolute error

are

⎧

⎪

⎨

⎪

⎩



i=1



(i) − Y

(i)





i=1

(i) − Y

(i)

Following from formulas (12.13)–(12.15), we realize that the ﬁtting error of

the average depth is very small. Therefore, we propose the following formula:

the sum of the depths of all the amino acids in a protein, −2.30257, is in

proportion to the square of the length of the protein. Also,



j=1

∼ 0.06580 × n

+2.30257 × n, (12.16)

in which n is the length of protein, and s

is the depth of jth amino acid. The

ﬁtting relationship of formula (12.11) is shown in Fig. 12.6.

12.2.3 Prediction for Depth of Multiple Peptide Chains

in Protein Spatial Structure

Using the protein depth database, we can predict the depth of each amino

acid in a protein. In Table 12.1, the relative frequency of the zero depth is

12.2 Statistical Depth Analysis of Protein Spatial Particles 343

the probability of ﬁnding the amino acid on the surface of the protein. From

Table 12.1, we can see that the probability that glutamic acid (E) is on the

surface of a protein is 32.66%. To improve the accuracy of our forecasts, we

can use multiple polypeptide chains on the surface of a protein.

Prediction of Tripeptide Chain

Let ABC be three linked amino acids. If one of them is on the surface of a pro-

tein, then we say that tripeptide chain ABC is on the surface of the protein. Let

n(A, B, C) be the number of tripeptides ABC on the surface of proteins in the

PDB database, let n

(A, B, C) be the number of all tripeptides on the surface

of proteins in the PDB database. Then f

(A, B, C) = n

(A, B, C)/n(A, B, C)

is the zero depth tendency factor of the tri-peptide ABC on the surface of the

protein. Table 12.4 shows all the tripeptide chains whose zero depth tendency

factor satisﬁes f

(A, B, C) > 75%.

Each tripeptide chain (for example, RED) consists of three amino acids.

Frequency is the number of of tripeptides in the entire amino acid sequence

whose length is greater than or at least equal to 50 in the PDB database, and

the zero depth tendency factor has the value of f

(A, B, C).

Prediction of Tetrapeptide

The deﬁnition of the zero depth tendency factor of tetrapeptide ABCD on

the surface of a protein is similar to the deﬁnition for a tripeptide. We list in

Table 12.5 the tetrapeptides whose zero depth tendency factors are greater

than 95%.

Following from Tables 12.4 and 12.5, we can get the probability of both

tripeptides and tetrapeptides appearing on the surface of the protein. For

example, from Table 12.5, we realize that tetrapeptide VYQR appears on the

surface of the protein with probability 97.142%. Therefore, we can determine

that the tetrapeptide VYQR must appear on the surface of the protein.

Remark 21. In Table 12.5, the zero depth tendency factor of a tetrapeptide

has been given under two conditions, namely, the zero depth tendency factor

of tetra-peptide must be more than 95%, and their frequency in the PDB

database must be more than 200. We list those tetrapeptides whose zero

depth tendency factors is more than 95%, with no restriction on frequency

in [99].

We can also calculate the zero depth tendency factor of polypeptide chains in

the PDB-Select database, for this may exclude the inﬂuence of homologous

sequences. Since the result is similar, we do not list these.