Biomathematics: Mathematics of Biostructures and Biodynamics

Front Cover
S. Andersson, K. Larsson, M. Larsson, M. Jacob
Elsevier, Oct 21, 1999 - Mathematics - 525 pages
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.
 

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Contents

Chapter 1 Introduction
1
Chapter 2 Counting Algebra and Periodicity the Roots of Mathematics are the Roots of Life
7
Chapter 3 Nodal Surfaces of Tetragonal and Hexagonal Symmetry and Rods
27
Chapter 4 Nodal Surfaces Planes Rods and Transformations
47
Chapter 5 Motion in Biology
73
Chapter 6 Periodicity in Biology Periodic Motion
105
Chapter 7 Finite Periodicity and the Cubosomes
131
Chapter 8 Cubic Cell Membrane SystemsCell Organelles and Periodically Curved Single Membranes
163
Chapter 14 The Lung Surface Structure and Respiration
341
Chapter 15 Epilogue
363
Appendix 1 The Plane the Cylinder and the Sphere
375
Appendix 2 Periodicity
385
Appendix 3 The Exponential Scale the GD function Cylinder and Sphere Fusion
399
Appendix 4 The Exponential Scale the Planes and the Natural Function Addition and Subtraction 409
409
Appendix 5 Multiplication of Planes Saddles and Spirals
419
Appendix 6 Symmetry
431

Chapter 9 Cells and their Division Motion in Muscles and in DNA
193
Chapter 10 Concentration Gradients Filaments Motor Proteins and again Flagella
223
Chapter 11 Transportation
257
Chapter 12 Icosahedral Symmetry Clathrin Structures Spikes Axons the Tree and Solitary Waves
285
Chapter 13 Axon Membranes and Synapses A Role of Lipid Bilayer Structure in Nerve Signals
313
Appendix 7 The Complex Exponential the Natural Exponential and the GD Exponential General Examples and Finite Periodicity
447
Appendix 8 Classical Differential Geometry and the Exponential Scale
463
Appendix 9 Mathematica Contains the Mathematica scripts used for calculating the equations for the figures in this book
477
Subject Index
521
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About the author (1999)

Kare Larsson, Camurus Lipid Research Foundation, Lund, Sweden.

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