Biomathematics

by Andersson; Larsson; Larsson; Jacob

• ISBN13: 9780444502735
• ISBN10: 0444502734
• Edition: 1st
• Format: Hardcover
• Copyright: 1999-10-21
• Publisher: Elsevier Science

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