Mathematical Biophysics, 2014
Biological and Medical Physics, Biomedical Engineering Series

This book presents concise descriptions and analysis of the classical and modern models used in mathematical biophysics. The authors ask the question "what new information can be provided by the models that cannot be obtained directly from experimental data?" Actively developing fields such as regulatory mechanisms in cells and subcellular systems and electron transport and energy transport in membranes are addressed together with more classical topics such as metabolic processes, nerve conduction and heart activity, chemical kinetics, population dynamics, and photosynthesis. The main approach is to describe biological processes using different mathematical approaches necessary to reveal characteristic features and properties of simulated systems. With the emergence of powerful mathematics software packages such as MAPLE, Mathematica, Mathcad, and MatLab, these methodologies are now accessible to a wide audience.

Preface

Part I Basic models in mathematical biophysics

Chapter 1 Growth and catalysis models
Unlimited growth. Exponential growth. Self-catalysis (Auto-catalysis)
Limited growth. The Verhulst equation
Constraints with respect to substrate. Models of Monod and Michaelis–Menten
Competition. Selection
Jacob and Monod trigger system
Classic Lotka and Volterra models
Models of species interactions
Models of the enzyme catalysis
Model of a continuous microorganism culture
Age structured populations
Leslie matrices
Continuous models of age structure

Chapter 2 Oscillations, rhythms and chaos in biological systems
Oscillations in glycolysis
Intracellular calcium oscillations
Deterministic Chaos
Chaos in the community of three species
Periodic supply of substrate in the system of glycolysis

Chapter 3 Spatiotemporal self-organization of biological systems
Waves of life
Autowaves and dissipative structures
Basic model “Brusselator”
Localized dissipative structures
Belousov–Zhabotinsky reaction

Chapter 4 Model of the impact of a weak electric field on the nonlinear system of trans-membrane ion transport
Transmembrane ion transport model
Bistable model
Auto –oscillating system

Part II Models of complex systems

Chapter 5 Oscillations and periodic space structures of pH and electric potential along the cell membrane of algae Chara corallina
Kinetic model of the proton ATPase (pump)
Equation, describing dynamics of proton concentration in the vicinity of the cell
Equation for potential dynamics
Oscillations in the local system
pH patterns along the cellular membrane
Dependence of the processes on light intensity. Hysteresis
Scheme of interactions of photosynthesis and ion fluxes leading to the nonlinear dynamics

Chapter 6 Models of Morphogenesis
Turing instability
Morphogenetic field
Model of a distributed trigger
Animal coat markings
Models of amoeba aggregation. The role of chemotaxis

Chapter 7 Autowave processes, nerve pulse propagation, and heart activity
Experiments and model of Hodgkin and Huxley
Reduced FitzHugh-Nagumo Model
Excited element of the local system
Running pulses
Detailed models of cardiomyocytes
Axiomatic models of excited medium. Autowave processes and cardiac arrhythmia

Chapter 8 Nonlinear models of DNA dynamics
Hierarchy of structural and dynamical models
Linear DNA theory
Simple linear model of an elastic bar
Nonlinear models of DNA mobility. Mechanical analogue
Mathematical model, simulating single DNA base’s nonlinear oscillations
Physical analogues of real DNA sequences
Long-range effects
Nonlinear mechanisms of transcription regulation

Part III Kinetic models of photosynthetic processes

Chapter 9 Models of photosynthetic electron transport. Electron transfer in a multienzyme complex
Organization of processes in photosynthetic membrane
Kinetic description of redox reactions in solution
Modeling electron transfer in a multienzyme complex
Electron transfer in a two-component complex
Electron transfer in a n-carrier complex
Electron transport via mobile carriers
Electron transport in an isolated photosynthetic reaction center

Chapter 10 Kinetic model of interaction of two photosystems
Types of regulation of photosynthetic processes
Model of PSI and PSII interaction
Subsystem PSII
Scheme of PSII states
Charge separation
Submodel of PSI
Description of the mobile carrier redox evolution
Relationships between total concentrations of electron carriers
Modeling of electron transport chain of wild type and mutant Arabidopsys thaliana

Chapter 11 Detailed model of electron transfer in PSIIFluorescence as an indicator of the state of the photosystem
Scheme of PSII states
Equations describing processes in PSII
Dependence of rate constants on thylakoid transmembrane electric potential
Energy loss processes
Experiment
Description of events in PSII electron transport system after a short light flash

Chapter 12 Generalized kinetic model of primary photosynthetic processes
The structure of the model
Photosystem II complex
Cytochrome b6f complex
Photosystem I complex
Mobile carriers in the kinetic model
Role of transmembrane electric potential
Transmembrane ion transfer and   generation
Buffer properties of lumen and stroma
Parameter values
Simulation of fluorescence transients at different light intensities
The role of different states of photosystem II in fluorescence induction
Simulation of   kinetics

Part IV Direct multiparticle models of processes in subcellular systems

Chapter 13 Method of direct multiparticle simulation of protein interactions
Restricted diffusion of mobile electron carriers in photosynthetic membrane
Direct model scene
Brownian dynamics of mobile carriers
Simulation of cyclic electron transport around photosystem I

Chapter  14 Modeling of protein complex formation in solution with diffusion and electrostatic interactions
Steps of redox protein interactions
Model of protein-protein interaction in solution
Protein diffusion. Approximation with ellipsoids of revolution
Simulation of geometric shape of proteins and their collisions
Electrostatic interactions
Simulation of complex formation
Docking rate constant dependence on ionic strength of solution
Comparative analysis of the interaction of Pc with Cyt f and PSI reaction centers in higher plants and cyanobacteria. Role of electrostatics

Chapter 15 Modeling of protein interactions in photosynthetic membrane
Interaction of Pc with Cyt f in thylakoid lumen
Modeling of Pc -PSI interaction considering membrane surface charge and multienzyme complexes embedded in the membrane
Modeling of Pc interaction with cyt f and PSI considering membrane surface charge and multienzyme complexes embedded in the membrane

Chapter 16 Spaciotemporal evolution of electrochemical potential ΔμH+ in photosynthetic membrane
Modeling of proton transfer
Model of proton release into lumen
Model of lateral diffusion of protons
Proton flow through the ATP-synthase and ATP synthesis
Computer simulation of proton gradient evolution and ATP creation

Conclusion
References
Index

Provides succinct but authoritative coverage of a broad array of biophysical topics and models

Written by authors at Moscow State University with its strong tradition in mathematics and biophysics

Scope, coverage, and length make the book highly suitable for use in a one-semester course at the senior undergraduate/graduate level