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w w w . c r c p r e s s . c o m A ScieNce PUBLiSheRS BOOK
LIVING MATTER
Algebra of Molecules
LIVING MATTER
Algebra of Molecules
Valery V. Stcherbic
Taras Shevchenko National University
Kiev, Ukraine
Leonid P. Buchatsky
Taras Shevchenko National University
Kiev, Ukraine
p,
A SCIENCE PUBLISHERS BOOK
GL--Prelims with new title page.indd ii 4/25/2012 9:52:40 AM
GL--Prelims with new title page.indd ii 4/25/2012 9:52:40 AM
CRC Press
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Preface
Living matter is built out of “nonliving molecules” and obeys the laws
of physics and mathematics. In living organisms, activation of chemical
substances takes place: biomolecules possess complex structure and fulfi l
specifi ed functions.
Living matter is a highly organized matter. Biochemical reactions, which
maintain self-organization of living matter, occur within a limited range of
environmental parameters. These reactions occur due to enzymes. Highly
coordinated biological processes determine the development of organisms
from a single cell. Biochemical processes obey the laws, appropriate to both
the most simple and the most highly developed organisms. Unity of living
matter is in the property of self-reproduction. Its basis is DNA, the main
molecule of life. DNA codes and genes, completely determine the structure
of the living organism.
A mathematical basis for building biological structures and biochemical
processes may someday be found. The laws of physics and chemistry are
quite universal but among them there is no the law of biological activation
of nonliving molecules. The emergence of life from nonliving matter and the
evolution of living organisms are some of the most complicated questions
standing before humankind.
Theoretical physics is an exact science with a high number of
mathematical approaches and laws that describe the surrounding world.
In biology, there are many “exceptions to the rules”; even genetic code
has variations. High accuracy of biological processes is the basis for the
emergence of various types of symmetries observed in structural molecular
biology. Forms of symmetry are often concealed in a mathematical
formula and cannot be seen to the observer of biological structures. For
each symmetry, there is an adequate law that represents maintenance of
some physical value in time. Application of algebraic methods can reveal
essential features of biological processes that are diffi cult to derive from
differential equations.
vi Living Matter: Algebra of Molecules
This reasoning is the basis for using Clifford algebra to describe
structures and processes in living matter. We do not know why Clifford
algebra may be the reason for biological activation of nonliving molecules,
but numerous examples considered in this book, show that Clifford algebra
possesses universality for adequate description of both biological structures
and biochemical processes.
The simplicity of the emergence of Clifford algebra is amazing. Any
section in the sequence of the growing cell number is a set of Clifford
algebras. If a geometrical structure possesses mirror symmetry (e.g.,
butterfl y’s wings), it is a spinor group of some Clifford algebra. But
asymmetrical geometrical structure also is connected with some Clifford
algebra, which is not so obvious.
Concepts of Clifford algebra play the leading part in this book; less
attention has been paid to internal symmetries of the algebra itself, e.g.,
to spinor groups. Main attention is paid to those structural elements of
biomolecules that determine a specifi c Clifford algebra.
The authors hope that the book will be interesting to specialists in
theoretical biology, quantum genetics, biophysics and bioinformatics and
would appreciate any remarks and suggestions.
March 2015 Valery V. Stcherbic
Leonid P. Buchatsky
Contents
Preface v
1. DNA Structure and Clifford Algebra 1
DNA Replication 2
Clifford Algebra 3
DNA Replication and Clifford Algebra 4
Complementary DNA Pairs 9
Clifford Algebra of DNA Nucleotides 10
Klein-Gordon Equation 14
Structural equations of 2-deoxyribose 15
Structural equations of phosphates 16
Structural equations of complementary pairs 16
Structural equations of nucleotide triplets 18
Appendix 1.1: Mathematics of Clifford Algebra 19
Appendix 1.2: Hadamard Matrices of Clifford Algebras 24
Appendix 1.3: Clifford Algebra Cl(4, 2) 26
Appendix 1.4: Presentation of Clifford Algebra by RNA Molecules 28
Selected Bibliography 29
2. tRNA Molecule in Affi ne Space 31
Clifford Algebras of tRNA and Aminoacyl-tRNA Synthetases 33
Curvature Tensor of Four-Dimensional Riemann Space 35
Curvature Tensor and L-Function of tRNA 36
Hidden Symmetry of the Genetic Code 39
Quadratic form of the genetic code 41
Graph of genetic code states 43
Interpretation of genetic code graph 45
Degeneracy of the Genetic Code 48
Selected Bibliography 51
3. Fine-Structure Constant in Presentation of Biomolecules 55
Fine-Structure Constant and Protein Structure 56
The structure of amino acids of the genetic code 56
Interpretation of reversed fi ne-structure constant 61
viii Living Matter: Algebra of Molecules
Fine-Structure Constant and ATP Molecule 67
Connection of ATP with fractional part of the constant 1/C 70
Fine-Structure Constant and DNA Structure 70
Fine-Structure Constant and Cap-Structure of mRNA 72
Appendix 3.1: Clifford Algebras of Simple Chemical Compounds 74
Appendix 3.2: Algebraic Formulae of the Constant 1/C 75
Rosen formula 75
Wyler formula 76
Comentary 3.1 7
Comentary 3.2 79
Selected Bibliography 79
4. Cancer: Viruses, Attractors, Fractals 83
Polyomaviruses 85
Skew symmetry Т = 3 of polyomavirus protein capsid 87
Stochastic Henon Attractor 88
Parameters of genome and proteins of polyomavirus capsid 88
Parameters of Henon map for polyomavirus proteins 90
Henon map parameters for antigens 92
Henon atractor 95
Fractal Properties of T-Antigens and Protein Capsid of 98
Polyomaviruses
L-factor of large and smal T-antigens 9
L-factor of polyomavirus protein capsid 100
Malignant Tumor is a Gigantic Compaction of the Fractal 101
State Space
Function of compaction of fractal states 101
Resonances of equivalence for benign tumour 103
Condensation of fractal states 105
Clifford Algebra of Tumour Growth 105
The Future of Oncology 111
Selected Bibliography 112
5. Transformation of Genetic Information into Clifford 115
Algebra
RNA Splicing 116
Splicing phenomenon 16
Group of alternatives 17
Diagrams of introns 19
Alternative splicing 121
Discrete Symmetry of Globular Proteins 124
Modular group SL(2, 5) 125
Clifford Algebra of Globular Proteins 129
Appendix 5.1: Clifford Algebra of Small T-Antigen of SV40 Virus 130
Appendix 5.2: Clifford Algebras of Genetic Code Amino Acids 133
Selected Bibliography 136
Contents ix
S upplementary Materials 139
DNA coding of signature of Cliford algebra 139
DNA transcription and bundle of Clifford algebra 141
Buterfl y Spin(p, q)-structure 142
;
Centrosome Spin (p, p)-structure 144
Selected readings 147
