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2002)

Index

“Aﬃnity” of amino acids 401

Alignment distance 222

A-distance 222

Alignment sequences

generated by envelope of pairwise

sequences 233

Alignment space 12, 222

A-space 222

Amino acids

order orientation 401

Anﬁnsen principle 265

Bend 280

Bernoulli

experiment 38

process 38

strength of 38

Binomial distribution 38

BLAST 7

Boolean algebra 73

Channel 369

Channel network 370

Chirality value se e mirror value

Combinational analysis method 416

Compound renewal sequence 56

Compressed operator for sequence 73

Connected empty sphere 369

Convex

analysis

convex closure 329

polyhedron 359

convex closure 359, 362

convex hull 359, 362

convex region 359

Core

generated by envelope 231

modulus structure 231

Counting process 38

Crystalline proteins 263

de Bruijn–Good graph 416

Decomposition of regions 79

Depth cut plane 327

depth cut triangle 328

Depth tendency factor 336

absolute zero 336

deepest 336

relative zero 336

Directed polyhedron 374

convex 374

directed hypertrophy 374

directed plane 374

directed triangle 374

envelope 374

retracting point 375

retracting tetrahedron 375

retracting triangle 375

retraction 375

simple envelope 375

simple retraction 375

Directed triangle 371

Distance 152

Dual renewal process 39

Dual renewal process of Poisson process

Dynamic programming-based algorithm

8, 17

442 Index

global alignment algorithm 23

Needleman–Wunsch algorithm 9, 18

Smith–Waterman algorithm 9, 22

traceback algorithm 19, 22

Envelope

generated by alignment sequences

235

generated by core 232

permutation 227

standard structure mode 227

structure mode 227

Equivalent representation of modes 70

Euclidean space 161

K-dimensional 358

metrical property 161

Expansion of ﬁnite set 14

Exponential distribution 41

Fast MA

CLUSTAL-W algorithm 173

FA S TA 7

Finite distance space 152

Finite metric space 152

Four-atom points

phase 283

type E 284

type Z 284

Four-dimensional protein structure

263

γ-accessible radius 377

Gene

crossing 7

positioning 7

repeating 7

searching 7

Genome

splicing 8

Geometric distribution 39

Hypergraph 359

arc or hyperarc 359

decomposition of hypergraph 360

order of hyperarc 359

order of hypergraph 359

point 359

i.i.d. sequences 31

Independent increment process 38

Independent stochastic sequence pairs

Information-based semantics analysis

397

Inserting positions of expanded

sequences 70

Interior cavity of protein 366

Interval

closed interval 25

half-open interval 25

open interval 25

Invertible property 74

Kullback–Leibler entropy 400

Length

deleted segment 60

inserted segment 60

inserted vector 69

permutated segment 54

Levenshtein distance 223

L-distance 223

Ligand and receptor 355

Limit theorem

almost everywhere convergence 36

central limit theorem 37

large number law 36

Linear and nonlinear complexity 420

Local autocorrelation function 25

Local correlation function 25

Local modiﬁcation operation 121

Local penalty function 36

Local vector 25

Local word 402

Locally random

autocorrelation function 33

correlation function 33

Markov property

tripeptide chain sequences 270

Maximum core and minimum envelope

uniqueness 230

Measurement 152

Metric space 221

metric expansion space 221

metric subspace 221

Mirror value 281

Mixed stochastic models

Index 443

parameter set 62

Modes

from multiple alignments 76

alternative representations 78

of pairwise alignments 78

Modulus structure

from multiple alignments 76

of multiple alignments 77

operations

associative law for intersection and

union 73

commutative law for intersection

and union 73

distributive law for intersection and

union 73

intersection for expanded modes

order relation in set of modes 73

recursion of order relation 73

reﬂexivity of order relation 73

subtraction of the expanded modes

union of the expanded modes 72

Modulus structure theory

compressed modulus structure 71

expanded mode 69

expanded modulus structure 69

Multiple alignment 10

optimization criteria 156

Multiple sequence

pairwise alignment 149

Mutated position 54

Mutated sequence

semistochastic model 64

type III 59

type IV 59

Mutation

displacement mutation 13

error mutation 13

ﬂow of 38

in biological sequences 7

stochastic analysis 29

type I 13, 45

type II 13, 50

model 51

type III 13

initial data 89

inserted data template 89

original template 89

type IV 13

Mutation point 39

Negative binomial distribution 39

Operator induced by expanded mode

Optimal alignment 16

Optimization criteria of MA

information-based criterion 166

rate of virtual symbols 172

similarity ratio 171

SP-penalty function 159

uniform alignment of multiple

sequences 157

weighted SP-function 159

Optimizing principle

average optimal rate 16

BLOSUM matrices 17

diﬀerence degree 15

Hamming matrix 15

optimizing function 15

optimizing matrix 15

PAM matrices 17

penalty matrix 15

scoring matrix 15

similarity 15

SP-function for multiple alignments

WT-matrix 15

Pairwise alignment

multiple sequences 149

fundamental theorem of minimum

penalty alignment 152

matrices 150

penalty matrix 151

within multiple sequences 25

pairwise alignment matrices 25

penalty matrices for pairwise

alignment 25

Parameter estimation of mutation mode

110

Phylogenetic tree 183

Bayes method 191

maximum likelihood 191

maximum parsimony 188

neighbor-joining method 185

UPGMA 184

444 Index

Poisson ﬂow 40

mixed 41

Poisson distribution 40

Poisson process 40

strength of 40

Prediction of homologous proteins 428

Prediction of peptide chain

surface of protein 343

Probability distribution family 30

Kolmogorov theorem 31

Probability distributions

measure of the diﬀerence between

400

Protein primary structure

grammar rules 395

Protein primary–secondary structure

probability distribution

status vector of tripeptide chain

268

sequence 268

status vector

of tripeptide chain 268

table of conditional entropy 269

Protein three-dimensional structure

measurement index of the homology

310

torsion angle sequence 308

alignment 308

consecutive alignment 309

discrete alignment 309

Purely shifting mutation 87

Random additive noise 47

Random interference 47

Relative entropy 400

density function 400

Renewal process 38

Representation for biological sequences

lengths 5

multisequence 6

representation for ﬁnite set 5

Sample track function

of counting process 42

of Poisson process 42

Scoring matrix

BLOSUM scoring matrix series 306

PAM scoring matrix series 306

Sequence

homologous sequence 13

local sequence 24

perfectly stochastic sequence 32

relationship between two

decomposition of sequence 68

expanded sequences 68

impressed sequences 68

maximal core 68

minimal envelope 68

real compression 68

real expansion 68

similar sequence 14

stochastic 29

continuously stochastic processes

inﬁnitely stochastic 30

random variable 29

random vector 29

representation of 30

stochastic processes 29

virtual expansion sequence 14

Sequential estimation 110

Set of positions 24

Set of subscripts 24

Shifting function 69

Shifting mutation 86

binary operations on the modes

intersection 94

order relationship 94

subtraction 94

symmetric subtraction 94

union 94

converse operator 97

decomposition of 87

deleting region 88

inserting region 88

nonshifting mutation region 88

decomposition of modulus structure

modulus structure 88

operators 96

position 88

representation of the modulus

structure 91

type and length 88

SMA

basic steps 175

comparison 179

Index 445

SPA 109

Space domain

closed region 358

Space polyhedron

boundary triangle 359

edge 359

polygon 359

vertex 359

Space region

bounded region 358

nature region 358

nondegenerate region 358

Spatial particle system

natural coordinate system 361

Standard normal distribution 37

Statistical decision-based algorithm for

pairwise alignment 109

Stochastic model

mixed mutation 61

type-III mutated sequence 58

type-IV mutated sequence 58

Strength of mutation 43

Strongly symmetric matrix 32

Super multiple alignment

SMA 172

Tetrahedrons V

294

Topological structure

boundary line 358

boundary point 358

boundary surface 358

cluster point 358

connected region 358

interior point 358

Torsion angle 281

Triangle splicing belt 289

plane 290

Tukey depth 325

Uniform alignment 100

Uniform distribution 31

Uniform local alignment 110

Virtual expanded 69

Virtual symbol 14

Word

key word and core word in

a database 421