9780444502735

Biomathematics

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  • ISBN13:

    9780444502735

  • ISBN10:

    0444502734

  • Edition: 1st
  • Format: Hardcover
  • Copyright: 1999-10-21
  • Publisher: Elsevier Science
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Summary

This book presents new mathematics for the description of structure and dynamics in molecular and cellular biology. On an exponential scale it is possible to combine functions describing inner organisation, including finite periodicity, with functions for outside morphology into a complete definition of structure. This mathematics is particularly fruitful to apply at molecular and atomic distances. The structure descriptions can then be related to atomic and molecular forces and provide information on structural mechanisms. The calculations have been focussed on lipid membranes forming the surface layers of cell organelles. Calculated surfaces represent the mid-surface of the lipid bilayer. Membrane dynamics such as vesicle transport are described in this new language. Periodic membrane assemblies exhibit conformations based on the standing wave oscillations of the bilayer, considered to reflect the true dynamic nature of periodic membrane structures. As an illustration the structure of an endoplasmatic reticulum has been calculated. The transformation of such cell membrane assemblies into cubosomes seems to reflect a transition into vegetative states. The organisation of the lipid bilayer of nerve cells is analyzed, taking into account an earlier observed lipid bilayer phase transition associated with the depolarisation of the membrane. Evidence is given for a new structure of the alveolar surface, relating the mathematical surface defining the bilayer organisation to new experimental data. The surface layer is proposed to consist of a coherent phase, consisting of a lipid-protein bilayer curved according to a classical surface - the CLP surface. Without employing this new mathematics it would not be possible to give an analytical description of this structure and its deformation during the respiration cycle. In more general terms this mathematics is applied to the description of the structure and dynamic properties of motor proteins, cytoskeleton proteins, and RNA/DNA. On a macroscopic scale the motions of cilia, sperm and flagella are modelled. This mathematical description of biological structure and dynamics, biomathematics, also provides significant new information in order to understand the mechanisms governing shape of living organisms.

Table of Contents

Introduction
1(6)
References 1
4(3)
Counting, Algebra and Periodicity-the Roots of Mathematics are the Roots of Life
7(20)
Counting and Sine
7(2)
Three Dimensions; Planes and Surfaces, and Surface Growth
9(7)
The Growth of Nodal Surfaces-Molecules and Cubosomes
16(11)
References 2
26(1)
Nodal Surfaces of Tetragonal and Hexagonal Symmetry, and Rods
27(20)
Non Cubic Surfaces
27(1)
Tetragonal Nodal Surfaces and their Rod Structures
27(9)
Hexagonal Nodal Surfaces and their Rod Structures
36(11)
References 3
45(2)
Nodal Surfaces, Planes, Rods and Transformations
47(26)
Cubic Nodal Surfaces
47(3)
Nodal Surfaces and Planes
50(6)
Cubic Nodal Surfaces and Parallel Rods
56(12)
Transformations of Nodal Surfaces
68(5)
References 4
72(1)
Motion in Biology
73(32)
Background and Essential Functions
73(3)
The Control of Shape-the Natural Exponential or cosh in 3D
76(5)
The Gauss Distribution (GD) Function and Simple Motion
81(12)
More Motion in 3D
93(12)
References 5
102(3)
Periodicity in Biology-Periodic Motion
105(26)
The Hermite Function
105(6)
Flagella - Snake and Screw Motion
111(5)
Periodic Motion with Particles in 2D or 3D
116(11)
Periodic Motion with Rotation of Particles in 2D
127(4)
References 6
130(1)
Finite Periodicity and the Cubosomes
131(32)
Periodicity and the Hermite Function
131(2)
Cubosomes and the Circular Functions
133(6)
Cubosomes and the GD-Function - Finite Periodicity and Symmetry P
139(4)
Cubosomes and the GD-Function - Symmetry G
143(4)
Cubosomes and the GD Function - Symmetry D
147(5)
Cubosomes and the Handmade Function
152(11)
References 7
162(1)
Cubic Cell Membrane Systems/Cell Organelles and Periodically Curved Single Membranes
163(30)
Introduction
163(1)
Cubic Membranes
163(6)
The Endoplasmatic Reticulum
169(6)
Protein Crystallisation in Cubic Lipid Bilayer Phases and Cubosomes - Colloidal Dispersions of Cubic Phases
175(2)
From a Minimal Surface Description to a Standing Wave Dynamic Model of Cubic Membranes
177(6)
Periodical Curvature in Single Membranes
183(10)
References 8
190(3)
Cells and their Division - Motion in Muscles and in DNA
193(30)
The Roots and Simple Cell Division
193(8)
Cell Division with Double Membranes
201(5)
Motion in Muscle Cells
206(7)
RNA and DNA Modelling
213(10)
References 9
220(3)
Concentration Gradients, Filaments, Motor Proteins and again - Flagella
223(34)
Background and Essential Functions
223(4)
Filaments
227(8)
Microtubulus and Axonemes
235(9)
Motor Proteins and the Power Stroke
244(3)
Algebraic Roots Give Curvature to Flagella
247(10)
References 10
255(2)
Transportation
257(28)
Background - Examples of Docking and Budding with Single Plane Layers, and Other Simple Examples
257(8)
Docking and Budding with Curved Single Layers
265(8)
Transport Through Double Layers
273(12)
References 11
284(1)
Icosahedral Symmetry, Clathrin Structures, Spikes, Axons, the Tree, and Solitary Waves
285(28)
The icosahedral symmetry
285(9)
Hyperbolic Polyhedra, Long Cones, Cylinders and Catenoids
294(5)
Cylinder Division and Cylinder Fusion - Cylinder Growth
299(6)
Solitary Waves, Solitons and Finite Periodicity
305(8)
References 12
311(2)
Axon Membranes and Synapses - A Role of Lipid Bilayer Structure in Nerve Signals
313(28)
The Nerve Impulse
313(2)
At the Action Potential Region of the Membrane there is a Phase Transition in the Lipid Bilayer
315(2)
A Model of a Phase-Transition/Electric Signal Coupling at Depolarisation and its Physiological Significance
317(10)
Transmission of the Nerve Signal at the Terminal Membrane of the Neurons - Synaptic Transmission
327(6)
Synchronisation of Muscle Cell Activation
333(2)
The General Anaesthetic Effect
335(2)
Physiological Significance of Involvement of a Lipid Bilayer Phase Transition in Nerve Signal Conduction
337(4)
References 13
338(3)
The Lung Surface Structure and Respiration
341(22)
The Alveolar Surface
341(1)
Lung Surfactant
342(2)
Structure of Tubular Myelin - A Bilayer arranged as the Classical CLP-Surface
344(5)
The Existence of a Coherent Surface Phase Lining the Alveoli
349(8)
Respiration
357(2)
Physiological Significance of the Existence of an Organised Surface Phase at the Alveolar Surface
359(4)
References 14
361(2)
Epilogue
363(12)
Acknowledgement
372(1)
References 15
372(3)
Appendix 1 The Plane, the Cylinder and the Sphere 375(10)
Appendix 2 Periodicity 385(14)
Appendix 3 The Exponential Scale, the GD function, Cylinder and Sphere Fusion 399(10)
Appendix 4 The Exponential Scale, the Planes and the Natural Function, Addition and Subtraction 409(10)
Appendix 5 Multiplication of Planes, Saddles and Spirals 419(12)
Appendix 6 Symmetry 431(16)
Appendix 7 The Complex Exponential, the Natural Exponential and the GD-Exponential - General Examples and Finite Periodicity 447(16)
Appendix 8 Classical Differential Geometry and the Exponential Scale 463(14)
Appendix 9 Mathematica (Contains the Mathematica scripts used for calculating the equations for the figures in this book.) 477

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