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**Theory and Mathematical Methods for Bioformatics ** written by
** Shiyi Shen** Nankai University, College of Mathematical Sciences, Tianjin, China and
**Jack A. Tuszynski**, University of Alberta, Department of Physics, Edmonton T6G 2G7, Alberta, Canada.
Bioinformatics is an interdisciplinary science which involves molecular biology,
molecular chemistry, physics, mathematics, computational sciences, etc.
Most of the books on biomathematics published within the past ten years have
consisted of collections of standard bioinformatics problems and informational
methods, and focus mainly on the logistics of implementing and making use of
various websites, databases, software packages and serving platforms. While
these types of books do introduce some mathematical and computational
methods alongside the software packages, they are lacking in a systematic
and professional treatment of the mathematics behind these methods.
It is significant in the field of bioinformatics that not only is the amount
of data increasing exponentially, but collaboration is also both widening and
deepening among biologists, chemists, physicists, mathematicians, and computer
scientists. The sheer volume of problems and databases requires researchers
to continually develop software packages in order to process the
huge amounts of data, utilizing the latest mathematical methods. The intent
of this book is to provide a professional and in-depth treatment of the
mathematical topics necessary in the study of bioinformatics.
This is an other great mathematics book cover the following topics.

Mutation and Alignment, Classification of Biological Sequences, Definition of Mutations and Alignments, Progress on Alignment Algorithms and Problems to Be Solved, Mathematical Problems Driven by Alignment and Structural Analysis, Basic Concepts in Alignment and Mathematical Models, Mutation and Alignment, Dynamic Programming-Based Algorithms for Pairwise Alignment, Introduction to Dynamic Programming-Based Algorithms, The Needleman–Wunsch Algorithm, the Global Alignment Algorithm, The Smith–Waterman Algorithm, Other Notations, Correlation Functions of Local Sequences, Pairwise Alignment Matrices Among Multiple Sequences, Remarks, Exercises, Analyses, and Computation

Stochastic Sequences and Independent Sequence Pairs, Definitions and Notations of Stochastic Sequences, Independently and Identically Distributed Sequences, Independent Stochastic Sequence Pairs, Local Penalty Function and Limit Properties of 2-Dimensional Stochastic Sequences, Stochastic Models of Flow Raised by Sequence Mutations, Bernoulli Processes, Poisson Flow, Mutated Flows Resulting from the Four Mutation Types, Stochastic Models of Type-I Mutated Sequences, Description of Type-I Mutation, Properties of Type-I Mutations, Type-II Mutated Sequences, Description of Type-II Mutated Sequences, Stochastic Models of Type-II Mutated Sequences, Error Analysis of Type-II Mutated Sequences, The Mixed Stochastic Models Caused by Type-I and Type-II Mutations, Mutated Sequences Resulting from Type-III and Type-IV Mutations, Stochastic Models of Type-III and Type-IV Mutated Sequences, Estimation of the Errors Caused by Type-III and Type-IV Mutations, Stochastic Models of Mixed Mutations, Exercises

Modulus Structure of Expanded and Compressed Sequences, The Modulus Structures of Expanded Sequences and Compressed Sequences, The Order Relation and the Binary Operators on the Set of Expanded Modes, Operators Induced by Modes, Modulus Structure of Sequence Alignment, Modulus Structures Resulting from Multiple Alignment, Structure Analysis of Pairwise Alignment, Properties of Pairwise Alignment, The Order Relation and the Operator Induced by Modulus Structure, Analysis of Modulus Structures Resulting from Sequence Mutations, Mixed Sequence Mutations, Structure Analysis of Purely Shifting Mutations, Structural Representation of Mixed Mutation, The Binary Operations of Sequence Mutations, The Order Relationship Among the Modes of Shifting Mutations, Operators Induced by Modes of Shifting Mutations, Error Analysis for Pairwise Alignment, Uniform Alignment of Mutation Sequences, Optimal Alignment and Uniform Alignment, Error Analysis of Uniform Alignment, Local Modification of Sequence Alignment, Exercises

Principle of Statistical Decision-Based Algorithms for Pairwise Sequences, Uniform Alignment and Parameter Estimation for Pairwise Sequences, The Locally Uniform Alignment Resulting from Local Mutation, The Estimations of Mutation Position and Length, Operation Steps of the SPA and Its Improvement, Operation Steps of the SPA, Some Unsolved Problems and Discussions of SPA, Local Modifications for Sequence Alignment, Index Analysis of SPA, The Statistical Features of Estimations, Improvement of the Algorithm to Estimate ˆs∗, The Computational Complexity of the SPA, Estimation for the Error of Uniform Alignment Induced by a Hybrid Stochastic Mutation Sequence, Applications of Sequence Alignment and Examples, Several Applications of Sequence Alignment, Examples of Pairwise Alignment, Exercises, Multiple Sequence Alignment, Pairwise Alignment Among Multiple Sequences, Using Pairwise Alignment to Process Multiple Sequences, Topological Space Induced by Pairwise Alignment of Multiple Sequences

The Definition of MA, Uniform Alignment Criteria and SP-Optimization Criteria for Multiple Sequences, Discussion of the Optimization Criterion of MA, Optimization Problem Based on Shannon Entropy, The Similarity Rate and the Rate of Virtual Symbols, Super Multiple Alignment, The Situation for MA, Algorithm of SMA, Comparison Among Several Algorithms, Exercises, Analyses, and Computation . . . . . . . . . . . . . . . . . . . . . 180

General Method of Constructing the Phylogenetic Tree, Summary, Distance-Based Methods, Feature-Based (Maximum Parsimony) Methods, Maximum-Likelihood Method and the Bayes Method, Network Structure Generated by MA, Graph and Tree Generated by MA, Network System Generated by Mutations of Multiple Sequences, The Application of Mutation Network Analysis, Selection of the Data Sample, The Basic Steps to Analyze the Sequences, Remarks on the Alignment and Output Analysis, Exercises, Analyses, and Computation

Construction of Alignment Space and Its Basic Theorems, What Is Alignment Space?, The Alignment Space Under General Metric, The Analysis of Data Structures in Alignment Spaces, Maximum Score Alignment and Minimum Penalty Alignment, The Structure Mode of the Envelope of Pairwise Sequences, Uniqueness of the Maximum Core and Minimum Envelope of Pairwise Sequences, The Envelope and Core of Pairwise Sequences, The Envelope of Pairwise Sequences and Its Alignment Sequences, The Counting Theorem of the Optimal Alignment and Alignment Spheroid, The Counting Theorem of the Optimal Alignment, Alignment Spheroid, The Virtual Symbol Operation in the Alignment Space, The Definition of the Virtual Symbol Operator, The Modulus Structure of the Virtual Symbol Operator, The Isometric Operation and Micro-Adapted Operation of Virtual Symbols, Exercises, Analyses, and Computation

Amino Acids and Peptide Chains, Amino Acids, Basic Structure of Peptide Chains, Brief Introduction of Protein Configuration Analysis, Protein Structure Database, Brief Introduction to Protein Structure Analysis, Analysis and Exploration

Protein Secondary Structure Prediction and Informational and Statistical Iterative Analysis, Protein Secondary Structure Prediction, Data Source and Informational Statistical Model of PSSP, Informational and Statistical Characteristic Analysis on Protein Secondary Structure, Informational and Statistical Calculation Algorithms for PSSP, Informational and Statistical Calculation for PSSP, Informational and Statistical Calculation Algorithm for PSSP, Discussion of the Results, Exercises, Analyses, and Computation

Space Conformation Theory of Four-Atom Points, Conformation Parameter System of Four-Atom Space Points, Phase Analysis on Four-Atom Space Points, Four-Atom Construction of Protein 3D Structure, Triangle Splicing Structure of Protein Backbones, Triangle Splicing Belts of Protein Backbones, Characteristic Analysis of the Protein Backbone Triangle Splicing Belts, Structure Analysis of Protein Side Chains, The Setting of Oxygen Atom O and Atom CB on the Backbones, Statistical Analysis of the Structures of Tetrahedrons VO, VB, Exercises, Analyses, and Computation

Structure Analysis for Protein Sequences, Alignment of Protein Sequences, Differences and Similarities Between the Alignment of Protein Sequences and of DNA Sequences, The Penalty Functions for the Alignment of Protein Sequences, Key Points of the Alignment Algorithms of Protein Sequences, Alignment of Protein Three-Dimensional Structures, Basic Idea and Method of the Alignment of Three-Dimensional Structures, Example of Computation in the Discrete Case, Example of Computation in Consecutive Case, Exercises, Analyses, and Computation

Depth Analysis of Amino Acids in Proteins, Introduction to the Concept of Depth, Definition and Calculation of Depth, Proof of Theorem, Proof of Theorem, Statistical Depth Analysis of Protein Spatial Particles, Calculation for Depth Tendency Factor of Amino Acid, Analysis of Depth Tendency Factor of Amino Acid, Prediction for Depth of Multiple Peptide Chains in Protein Spatial Structure, The Level Function in Spatial Particle System, An Example, Exercises

Introduction, Morphological Features of Protein Spatial Structure, Several Basic Definitions and Symbols, Preliminary Analysis of the Morphology of Spatial Particle System , Example, Structural Analysis of Cavities and Channels in a Particle System, Definition, Classification and Calculation of Cavity, Relationship Between Cavities, Example, Analysis of γ-Accessible Radius in Spatial Particle System, Structural Analysis of a Directed Polyhedron, Definition and Basic Properties of γ-Accessible Radius, Basic Principles and Methods of γ-Accessible Radius, Recursive Algorithm of γ-Function, Calculation of the γ-Function Generated by 0-Level Convex Hull, Recursive Calculation of γ-Function, Example, Proof of Relative Theorems and Reasoning of Computational Formulas, Proofs of Several Theorems, Reasoning of Several Formulas, Exercises

Semantic Analysis for Protein Primary Structure, The Definition of Semantic Analysis for Protein Primary Structure, Information-Based Stochastic Models in Semantic Analysis, Determination of Local Words Using Informational and Statistical Means and the Relative Entropy Density Function for the Second and Third Ranked Words, Semantic Analysis for Protein Primary Structure, Permutation and Combination Methods for Semantic Structures, Notation Used in Combinatorial Graph Theory, The Complexity of Databases, Key Words and Core Words in a Database, Applications of Combinatorial Analysis, Exercises, Analyses, and Computation

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