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

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364 13 Morphological Features of Protein Spatial Structure

Table 13.2. (continued)

No. (i, j, k)No.(i, j, k)No.(i, j, k)No.(i, j, k)No.(i, j, k)

1 3,6,7 2 3,6,164 3 3,7,25 4 3,25,93 5 3,26,93

6 3,26,90 7 3,164,319 8 3,319,27 9 3,27,90 10 6,7,72

11 6,72,164 12 7,25,148 13 7,72,148 14 25,93,148 15 26,93,147

16 26,147,37 17 26,36,37 18 26,36,90 19 51,52,71 20 51,52,75

21 51,71,147 22 51,147,37 23 51,37,75 24 52,53,70 25 52,53,147

26 52,70,69 27 52,71,147 28 52,69,74 29 52,74,75 30 53,70,288

31 53,147,148 32 53,148,288 33 70,288,57 34 70,57,69 35 72,73,162

36 72,73,164 37 72,148,160 38 72,160,162 39 73,162,164 40 93,147,148

41 148,160,182 42 148,182,202 43 148,202,297 44 148,288,297 45 160,162,182

46 162,164,207 47 162,182,207 48 164,172,183 49 164,172,207 50 164,183,187

51 164,187,319 52 172,183,208 53 172,198,208 54 172,198,223 55 172,207,223

56 182,202,207 57 183,187,208 58 187,208,265 59 187,209,218 60 187,209,267

61 187,218,265 62 187,267,319 63 198,208,220 64 198,220,223 65 202,207,244

66 202,244,245 67 202,245,297 68 207,223,225 69 207,225,245 70 207,244,245

71 208,220,287 72 208,265,285 73 208,285,304 74 208,287,304 75 209,218,267

76 218,265,282 77 218,266,267 78 218,266,282 79 220,223,287 80 223,225,287

81 225,245,263 82 225,263,287 83 245,261,297 84 245,261,5 85 245,263,5

86 261,297,5 87 263,287,303 88 263,303,5 89 265,282,285 90 266,267,273

91 266,273,317 92 266,282,315 93 266,315,317 94 267,273,319 95 273,317,319

96 282,285,304 97 282,304,8 98 282,315,8 99 287,303,304 100 288,297,5

101 288,5,55 102 288,55,57 103 303,304,5 104 304,5,8 105 315,317,4

106 315,4,9 107 315,8,9 108 317,319,321 109 317,321,19 110 317,4,19

111 319,320,24 112 319,320,27 113 319,321,23 114 319,23,24 115 320,24,27

116 321,19,23 117 4,9,19 118 5,8,57 119 5,55,57 120 8,9,69

121 8,57,69 122 9,19,69 123 19,23,54 124 19,54,69 125 23,24,54

126 24,27,37 127 24,37,50 128 24,50,54 129 27,36,37 130 27,36,90

131 37,50,75 132 50,54,74 133 50,74,75 134 54,69,74

1 3,6,70 2 3,6,73 3 3,53,147 4 3,53,57 5 3,70,147

6 3,73,208 7 3,208,23 8 3,23,57 9 6,70,71 10 6,71,72

11 6,72,73 12 7,25,52 13 7,25,54 14 7,52,55 15 7,54,55

16 25,26,51 17 25,26,50 18 25,51,160 19 25,52,160 20 25,24,50

21 25,24,54 22 26,51,160 23 26,160,304 24 26,304,37 25 26,36,37

26 26,36,50 27 52,53,148 28 52,53,57 29 52,148,160 30 52,55,57

31 53,147,148 32 70,71,160 33 70,147,160 34 71,72,160 35 72,73,93

36 72,93,160 37 73,93,162 38 73,162,164 39 73,164,208 40 93,160,187

41 93,162,187 42 147,148,160 43 160,187,225 44 160,223,225 45 160,223,315

46 160,304,315 47 162,164,182 48 162,182,198 49 162,183,187 50 162,183,198

51 164,172,182 52 164,172,208 53 172,182,198 54 172,198,202 55 172,202,245

56 172,207,208 57 172,207,261 58 172,245,261 59 183,187,225 60 183,198,244

61 183,225,244 62 198,202,220 63 198,220,244 64 202,220,245 65 207,208,209

66 207,209,218 67 207,218,261 68 208,209,263 69 208,263,285 70 208,285,23

71 209,218,263 72 218,261,266 73 218,263,266 74 220,244,267 75 220,245,267

76 223,225,244 77 223,244,267 78 223,267,317 79 223,315,317 80 245,261,273

81 245,267,273 82 261,266,273 83 263,265,266 84 263,265,285 85 265,266,288

86 265,282,285 87 265,282,288 88 266,273,288 89 267,273,303 90 267,303,317

13.1 Introduction 365

Table 13.2. (continued)

No. (i, j, k)No.(i, j, k)No.(i, j, k)No.(i, j, k)No.(i, j, k)

91 273,288,297 92 273,297,320 93 273,303,320 94 282,285,287 95 282,287,288

96 285,287,5 97 285,321,4 98 285,321,23 99 285,4,5 100 287,288,5

101 288,297,320 102 288,320,5 103 303,304,317 104 303,304,319 105 303,319,8

106 303,320,8 107 304,315,317 108 304,319,8 109 304,8,9 110 304,9,37

111 320,5,8 112 321,4,19 113 321,19,36 114 321,23,36 115 4,5,9

116 4,9,19 117 5,8,9 118 9,19,37 119 19,36,37 120 23,24,27

121 23,24,55 122 23,27,36 123 23,55,57 124 24,27,50 125 24,54,55

126 27,36,50

1 3,6,7 2 3,6,73 3 3,7,183 4 3,53,70 5 3,53,73

6 3,70,160 7 3,160,164 8 3,164,172 9 3,172,183 10 6,7,25

11 6,25,52 12 6,52,73 13 7,25,183 14 25,51,52 15 25,51,288

16 25,183,288 17 26,51,287 18 26,51,288 19 26,273,287 20 26,273,288

21 51,52,73 22 51,72,73 23 51,72,287 24 53,70,71 25 53,71,73

26 70,71,148 27 70,148,162 28 70,160,162 29 71,73,147 30 71,147,148

31 72,73,218 32 72,209,218 33 72,209,263 34 72,263,287 35 73,93,147

36 73,93,187 37 73,187,218 38 93,147,187 39 147,148,187 40 148,162,187

41 160,162,208 42 160,164,202 43 160,202,207 44 160,207,208 45 162,187,198

46 162,198,208 47 164,172,182 48 164,182,267 49 164,202,220 50 164,220,261

51 164,261,267 52 172,182,183 53 182,183,273 54 182,267,273 55 183,273,288

56 187,198,218 57 198,208,225 58 198,209,218 59 198,209,265 60 198,225,265

61 202,207,220 62 207,208,225 63 207,220,225 64 209,263,265 65 220,223,225

66 220,223,261 67 223,225,245 68 223,245,261 69 225,244,245 70 225,244,265

71 244,245,265 72 245,261,266 73 245,265,266 74 261,266,267 75 263,265,287

76 265,266,282 77 265,282,285 78 265,285,287 79 266,267,273 80 266,273,282

81 273,282,287 82 282,285,287

1 3,6,51 2 3,6,93 3 3,51,52 4 3,52,172 5 3,93,182

6 3,172,182 7 6,7,26 8 6,7,93 9 6,26,52 10 6,51,52

11 7,25,26 12 7,25,160 13 7,71,72 14 7,71,160 15 7,72,73

16 7,73,93 17 25,26,52 18 25,52,53 19 25,53,160 20 52,53,147

21 52,147,172 22 53,147,148 23 53,148,160 24 70,72,160 25 70,72,162

26 70,160,162 27 71,72,160 28 72,73,182 29 72,162,182 30 73,93,182

31 147,148,162 32 147,162,164 33 147,164,172 34 148,160,162 35 162,164,172

36 162,172,182

1 3,6,7 2 3,6,70 3 3,7,25 4 3,25,26 5 3,26,52

6 3,52,53 7 3,53,70 8 6,7,160 9 6,70,93 10 6,93,147

11 6,147,160 12 7,25,160 13 25,26,160 14 26,51,52 15 26,51,162

16 26,148,160 17 26,148,162 18 51,52,53 19 51,53,71 20 51,71,73

21 51,73,162 22 53,70,71 23 70,71,72 24 70,72,162 25 70,93,162

26 71,72,162 27 71,73,162 28 93,147,162 29 147,148,160 30 147,148,162

1 3,6,7 2 3,6,70 3 3,7,25 4 3,25,70 5 6,7,52

6 6,51,52 7 6,51,70 8 7,25,52 9 25,26,51 10 25,26,52

11 25,51,70 12 26,51,52

366 13 Morphological Features of Protein Spatial Structure

13.2 Structural Analysis of Cavities and Channels

in a Particle System

A cavity is one of the basic features of the morphology of a particle system,

which we will discuss ﬁrst. The main method of calculation is the rolling

sphere method.

13.2.1 Deﬁnition, Classiﬁcation and Calculation of Cavity

For a ﬁxed spatial particle system A, its convex closure is Ω(A). We denote

the sphere with center o and radius r by O(o, r), and denote a rolling sphere

with radius r by O(r).

Deﬁnition of Empty Sphere

Deﬁnition 44. 1. If O(r) ∩ Ω(A) is not an empty set, and O(r) does not

contain any points of A,thenO(r) is a hollow (or empty) sphere with

radius r in particle system A.

2. If a, b, c, d are four points in A, their circumscribed sphere is O(a, b, c, d),

If O(a, b, c, d) is an empty sphere in particle system A, then we call

O(a, b, c, d) an empty sphere generated by the four points a, b, c, d.

Classiﬁcation of Cavities

There are many methods for classifying the relationship between a cavity and

the particle system A. First, we present the following four types of relation-

ships between O(r)andΩ(A):

1. Sphere O(r

)isentirelyinΩ(A), we call it an I-0-class cavity of particle

system A. In an I-0 class cavity, the cavity with the largest radius is the

largest cavity in particle system A.

2. If sphere O(r)ispartlyinΩ(A), then sphere O(r) must intercept several

boundary surfaces of Ω(A). If there exists a boundary surface π that

intercepts most of the sphere O(r) on the same side of Ω(A), then we

call this cavity a I-1-class cavity of particle system A,otherwise,wecall

it a I-2-class cavity. I-1 and I-2 class cavities are also called spheroidal

grooves of particle system A.

3. In addition, we can provide a classiﬁcation using circumscribed sphere

O(a, b, c, d). Thus, the relationship between a, b, c, d ∈ A and set A

stated as follows: For a, b, c, d ∈ A, if there exist zero, one, two, three, or

four points in set A

, then the corresponding O(a, b, c, d)isaII-0,II-1,

II-2, II-3, or II-4 class sphere, respectively. Since we do not deﬁne the

concept of II-4 class spheres in this chapter, we only discuss the ﬁrst four

types of cavities. For a II-0 class cavity, we can deﬁne the largest cavity

of the particle system A similarly.

13.2 Structural Analysis of Cavities and Channels 367

4. If sphere O is both a I-1-class and a II-3-class sphere, then a boundary

triangle δ = δ(a, b, c)ofΩ mustbeonthesurfaceofthesphere.WecallO

aI-1(r)-class sphere, where r = r(δ) is the circumradius of the boundary

triangle δ.

Search and Calculation of a Cavity

Searching and calculating the cavity can be executed in two steps, namely:

1. For any four points a, b, c, d in set A, we can determine a circumscribed

sphere O(a, b, c, d) by the four-point-determined-sphere method, with the

corresponding formulas given in Sect. 13.5. We denote the center and

radius of the sphere by o = o(a, b, c, d), r = r(a, b, c, d), respectively.

2. Let A(a, b, c, d)=A −{a, b, c, d}, if for any e ∈ A(a, b, c, d), e is not in

sphere O(a, b, c, d), then O(a, b, c, d)isacavityofA.

3. For a protein with length n, the computational complexity of ﬁnding all

its cavities is n

. Hence it is easily computed. For a protein with length

400, it takes 5 h to compute all the cavities on a 1 GHz PC. If we develop

a proper algorithm, the computational complexity can be reduced. We do

not discuss this further here.

Remark 24. The left object and the right object in Fig. 13.3 are the spatial

structure of yeast hexokinase PII and E. coli OmpF porin, respectively. Each

small sphere represents an atom; the two larger spheres are the “pocket” and

the “center cavity.”

Fig. 13.3. A protein’s “cavity” and the “pockets” of the surface

368 13 Morphological Features of Protein Spatial Structure

13.2.2 Relationship Between Cavities

Connected Cavities

If two known cavities O(r), O



) intersect, then they are called connected

cavities. The intersected curve of the surfaces of connected cavities O(r),



) must be a circle. Below, we provide the following discussion.

1. Let o, o



be the centers of cavities O(r), O



), respectively. We de-

note the distance between the two centers by h = |oo



|, and denote

a point on the intersected curve of the surface of the cavities by a.

Point a



is the projection of point a on line oo



.Thus,a



divides the

line oo



into two parts: oa



, a



, whose lengths are x, y, respectively.

Their geometrical relationship and computational result are shown in

Fig. 13.4.

2. x, y satisfy equations:



x + y = h,

− x

=(r



)

− y

Solving these equations, we have that

⎧

⎪

⎨

⎪

⎩

x =

− (r



)

y =

−

− (r



)

(13.2)

Therefore, the values of x, y are independent of the selection of point a,

and we denote a



by o



Fig. 13.4. Computation map of the radius of section between connected spheres

13.2 Structural Analysis of Cavities and Channels 369

3. Let the intersected curve of the surface of O(r),O



)beL, then the

distance between any point a on L and o



is computed as follows:



=(r

− x

)

1/2

−



− (r



)



1/2



2[r

+(r



)

] − h

− 2r



)

+(r



)



1/2

. (13.3)

Obviously, the values of r



and point a are independent. So the intersected

curve of the surface of O(r), O



), which we denoted by L,isacircle

with center o



and radius r



. The corresponding formulas are presented

in (13.2) and (13.3). We call r



in (13.3) the radius of the section between

the two connected spheres.

Basic Properties of Connected Empty Spheres

If O(r

), O



) are two connected empty spheres of set A,andr



is their

radius of section, then we say that the two empty spheres form a sphere pair,

denoted by (r



). The following properties hold:

1. r



≤ min{r

2. For any sphere O(r) such that r<r



holds, the sphere can move freely

within the connected empty spheres without encountering any points of

set A.

3. If O(a, b, c, d), O



) are two connected empty spheres, then

a, b, c, d, a



cannot be interior points of region

O(a, b, c, d) ∪ O



) .

Connected Empty Spheres Map Generated by Particle System

For a ﬁxed particle system A, we denote its entire connected empty spheres

map by G

= {O

, V

(O,O



)

},where



= {O(a, b, c, d): a, b, c, d ∈ A, and O(a, b, c, d) is empty sphere of A} ,

= {(O, O



): O, O



∈Q

, and (O, O



) is connected sphere of A} ,

(13.4)

where r

is the radius of sphere O,andr

O,O



is the radius of the section

of connected empty spheres (O, O



). Then G

is the connected empty sphere

map generated by particle system A,whereO

contains all the vertices of

the map, and V

contains all the arcs of the map, while r

, r

(O,O



)

are the

coloring functions of the vertices and arcs, respectively.

370 13 Morphological Features of Protein Spatial Structure

Fig. 13.5. Channel network map

The Threshold Subgraph

The threshold subgraph can be determined by the connected empty spheres

map G

.Ifweletr

∗

be a ﬁxed constant, we deﬁne the threshold subgraph of

by G

∗

)={O

∗

), V

∗

),r

(O,O



)

},where



∗

)={O ∈O

: r

≥ r

∗

} ,

∗

)={(O, O



) ∈V

: r

O,O



≥ r

∗

} .

(13.5)

Therefore, in graph G

∗

), r

(O,O



)

≥ r

∗

always holds.

Deﬁnition 45. In the threshold subgraph of the connected empty sphere

∗

), the path is a channel in particle system A.

If L is a path in graph G

∗

), then for any sphere O(r) such that r<r



holds, the sphere can move freely through the path without encountering any

points of set A.

To analyze a channel, we use the same deﬁnitions and symbols as in graph

theory, such as the start vertex and end vertex, cycle, and so on. Several

special deﬁnitions are listed as follows:

1. Channel L is connected with the exterior of particle system A,ifthe

beginning sphere O of channel L is connected with another sphere O



the radius of sphere O



and the connected radius are r



≥ r

∗

,and

if sphere O



= O(δ) is a circumscribed sphere of a boundary triangle of

Ω(A). Then we also say that one end of the channel is connected with the

exterior of particle system A.

2. Channel L is penetrable according to particle system A, if both endpoints

of channel L are connected with the exterior of particle system A.

3. If two channels L

, L

have common points, then we say that they cross.

Several crossed channels compose a channel network. See Fig. 13.5 for an

illustration. Finding and characterizing a channel network is simply grid

computing in graph theory, which we will not discuss here.

13.3 Analysis of γ-Accessible Radius in Spatial Particle System 371

13.2.3 Example

We analyze the empty cavity and channel of protein E. coli OmpF porin. The

data of the spatial structure of C

atoms are given in Table 12.6. We continue

our discussion based on these data:

1. In particle system A, the total number of cavities is 2033, where the radii

range from 20.0to60.0, and the number of cavities whose four vertices

are all in Ω(A) is 42. These are shown in Table 13.3.

2. Among these 42 cavities, let G

∗

)={O

∗

), V

∗

),r

(O,O



)

} de-

note the channel network map constructed by the connected sphere pairs

such that the radius of section r

∗

> 20 holds, where O

∗

),r

are the

parameters given in Table 13.3, and V(r

∗

), r

O,O



are given in Table 13.4:

3. From Table 13.4, we can identify a series of channels as follows:

35 → 26 → 23 → 24 → 2 → 25 → 31 → 22 → 29

15 → 24 → 23 → 26 → 35 → 22 → 31 → 25 → 12 ,

and so on. A small sphere with radius r<20 can move through these

channels freely, without encountering any points in set A.

If these channels are connected with the circumscribed sphere of the

boundary surface, then they are penetratable in convex closure Ω(A).

We do not discuss this further at this point.

13.3 Analysis of γ-Accessible Radius in Spatial Particle

System

13.3.1 Structural Analysis of a Directed Polyhedron

In order to calculate and analyze spatial particle systems, we will ﬁrst in-

troduce the theory of the structure of a directed polyhedron. In Sect. 13.1,

we have introduced the hypergraph which is used to represent the spatial

polyhedron; now we introduce the structure of the directed polyhedron.

In diﬀerential geometry, the theory of exterior diﬀerentiation is used to

describe the direction of boundary surfaces of spatial regions. We do not use

the theory here, but instead focus on the structure of the polyhedron. We

explain using the structure of the directed polyhedron as follows.

Directed Triangle

The triangle δ which has been given a normal direction is a directed triangle,

denoted by δ. It divides the space into two sides: inside and outside. The

direction of δ of the boundary triangle on a convex polyhedron is deﬁned:

such that one side, in which set Ω lies, is the inside, and the other is the

outside.

372 13 Morphological Features of Protein Spatial Structure

Table 13.3. Data table of cavities of orotein E. coli OmpF porin (vertices in convex hull)

No. No. of 4 vertices X, Y, Z; R No. No. of 4 vertices X, Y, Z; R

1 4, 11, 13, 45 115.18, 32.45, 19.63; 20.54 2 4, 11, 45, 61 119.73, 30.42, 13.42; 28.04

3 4, 13, 44, 45 122.11, 26.09, 13.73; 31.34 4 4, 13, 44, 76 122.23, 25.96, 13.69; 31.50

5 4, 44, 45, 76 122.26, 26.01, 13.54; 31.58 6 4, 45, 61, 76 122.65, 26.15, 12.66; 32.26

7 9, 11, 59, 61 119.16, 33.88, 10.63; 28.08 8 9, 49, 59, 86 113.74, 42.72, 10.74; 21.32

9 13, 43, 44, 76 121.95, 25.89, 14.07; 31.10 10 21, 23, 36, 37 90.56, −1.11, 59.52; 29.57

11 21, 23, 334, 336 99.39, −15.77, 84.92; 59.45 12 43, 44, 61, 76 116.15, 26.83, 15.55; 25.49

13 43, 61, 63, 76 112.42, 27.25, 16.81; 21.86 14 44, 45, 61, 76 121.76, 26.22, 13.29; 31.24

15 59, 61, 84, 86 110.88, 35.85, 10.64; 21.22 16 61, 63, 76, 81 110.42, 27.58, 15.96; 20.51

17 61, 76, 81, 84 116.98, 27.51, 4.12; 31.87 18 61, 76, 84, 86 132.73, 26.16, −7.41; 50.87

19 76, 81, 84, 163 114.51, 26.82, −1.88; 34.12 20 81, 84, 99, 163 102.53, 30.54, 5.79; 20.33

21 84, 86, 99, 163 128.92, 25.99, −13.94; 52.18 22 86, 98, 99, 135 114.56, 42.74, −15.01; 40.44

23 88, 96, 98, 137 97.66, 56.69, −1.09; 22.53 24 88, 96, 137, 159 101.80, 71.54, −25.39; 50.32

25 88, 98, 137, 159 103.62, 60.16, −15.30; 37.95 26 144, 145, 155, 174 65.96, 80.47, −4.42; 39.16

27 144, 155, 174, 176 61.79, 74.35, 4.18; 32.12 28 144, 174, 176, 191 48.40, 82.52, −0.52; 47.19

29 144, 174, 191, 193 43.43, 87.07, −4.03; 54.65 30 144, 176, 178, 191 50.51, 79.98, 2.52; 42.90

31 173, 174, 195, 210 64.29, 65.08, −10.47; 32.56 32 189, 191, 210, 235 27.11, 79.97, 4.33; 57.59

33 189, 191, 214, 233 51.59, 58.96, 17.21; 23.43 34 189, 191, 233, 235 34.85, 72.24, 8.94; 45.97

35 189, 216, 231, 233 37.17, 65.04, 24.99; 36.92 36 189, 231, 233, 255 28.19, 70.63, 24.80; 47.39

37 191, 193, 210, 212 52.49, 60.41, 9.88; 25.65 38 191, 210, 212, 235 47.21, 62.85, 9.82; 30.96

39 191, 212, 233, 235 54.27, 55.35, 14.99; 20.03 40 216, 231, 232, 233 50.92, 55.71, 25.51; 20.55

41 233, 253, 255, 281 46.96, 49.57, 20.22; 20.41 42 301, 309, 311, 334 85.39, 9.94, 75.43; 34.40

The amino acids represented by the serial number of four vertices, as well as its coordinates, are shown in

Table 12.6. X, Y, Z; R the coordinates of the circumcenter and circumradius

13.3 Analysis of γ-Accessible Radius in Spatial Particle System 373

Table 13.4. Data table of connected spheres of protein E. coli OmpF porin

Connected

sphere



O,O



Connected

sphere



O,O



Connected

sphere



O,O



(2, 24) 28.04, 50.32, 23.73 (2, 25) 28.04, 37.95, 23.81 (3, 24) 31.34, 50.32, 24.60

(3, 25) 31.34, 37.95, 24.54 (4, 24) 31.50, 50.32, 24.69 (4, 25) 31.50, 37.95, 24.62

(5, 24) 31.58, 50.32, 24.82 (5, 25) 31.58, 37.95, 24.73 (6, 24) 32.26, 50.32, 25.69

(6, 25) 32.26, 37.95, 25.46 (7, 24) 28.08, 50.32, 25.58 (7, 25) 28.08, 37.95, 25.44

(9, 24) 31.10, 50.32, 24.23 (9, 25) 31.10, 37.95, 24.23 (12, 25) 25.49, 37.95, 20.49

(14, 24) 31.24, 50.32, 24.78 (14, 25) 31.24, 37.95, 24.70 (15, 24) 21.22, 50.32, 20.48

(17, 24) 31.87, 50.32, 28.72 (17, 25) 31.87, 37.95, 28.19 (18, 24) 50.87, 50.32, 41.53

(19, 23) 34.12, 22.53, 21.22 (19, 24) 34.12, 50.32, 31.58 (21, 24) 52.18, 50.32, 43.47

(22, 26) 40.44, 39.16, 24.68 (22, 29) 40.44, 54.65, 21.46 (22, 31) 40.44, 32.56, 23.65

(23, 26) 22.53, 39.16, 21.42 (23, 31) 22.53, 32.56, 20.15 (24, 28) 50.32, 47.19, 38.42

(24, 29) 50.32, 54.65, 41.48 (24, 30) 50.32, 42.90, 35.81 (24, 32) 50.32, 57.59, 35.61

(24, 34) 50.32, 45.97, 30.00 (24, 36) 50.32, 47.39, 20.06 (24, 37) 50.32, 25.65, 20.35

(24, 38) 50.32, 30.96, 22.97 (25, 26) 37.95, 39.16, 31.60 (25, 27) 37.95, 32.12, 25.21

(25, 28) 37.95, 47.19, 29.17 (25, 29) 37.95, 54.65, 31.00 (25, 30) 37.95, 42.90, 27.32

(25, 31) 37.95, 32.56, 28.79 (25

, 32) 37.95, 57.59, 24.25 (26, 35) 39.16, 36.92, 31.00

(26, 36) 39.16, 47.39, 35.24 (31, 35) 32.56, 36.92, 26.49 (31, 36) 32.56, 47.39, 29.53

The spheres represented by the serial numbers of connected spheres are given in Table 13.3, and



O,O



represent the radius and the connected radius, respectively