Nonlinear Dynamics of Nanobiophysics

Contents
Introduction
	References
The Insights into Richness of Nonlinear Schrödinger Equation
	1 Introduction
	2 Nonlinear Schrödinger Equation
		2.1 Modulation Instability
		2.2 Solitons
		2.3 Breathers
		2.4 Rogue Waves
	3 Generalized NLSE
		3.1 Higher-Order NLSE
		3.2 Driven NLSE with Quadratic-Cubic Nonlinearity
	4 Applications of Nonlinear Localized Waves in Biology
	5 Conclusion
	References
Nonlinear Dynamics of DNA Chain
	1 DNA Dynamics
	2 Resonance Mode and DNA Opening
	3 Demodulated Standing Solitary Wave and DNA–RNA Transcription
	References
Nonlinear Dynamics of DNA Chain  with Long-Range Interactions
	1 Introduction
	2 Long-Range Interactions in Biological Systems
		2.1 Long-Range Interactions of the Kac-Baker Type
		2.2 Long-Range Interactions of the Power-Law Type
		2.3 Physical Nature of Long-Range Interactions of the Power-Law Type
	3 Long-Range Interactions in the HPB Model
		3.1 Model Hamiltonian
		3.2 Equations of Motions
		3.3 Discrete Derivation Operator Technique
		3.4 Soliton Solutions
	4 Long-Range Interactions in the HPB Model  with Damping Effect
		4.1 Long-Range Hydrodynamical Damping Forces  and Equations of Motions
		4.2 Dissipative Soliton Solution
	5 Conclusion
	References
Trajectories of DNA Kinks
	1 Kinks of Homogeneous DNA
	2 Kink Trajectories in Homogeneous DNA
		2.1 Kink Trajectories in the Case of Absence of External Field
		2.2 Kink Trajectories in the Case of Constant External Field M0
		2.3 Kink Trajectories in the Case of Periodic External Field with Constant Frequency M( t ) = M0 cos(2Ωt)
		2.4 Kink Trajectories in the Case of Periodic External Field with Slowly Varying Frequency  M( t ) = M0 cos(Ωt - αt2 /2)
		2.5 Kink Trajectories in the Case of on/off External Field
	3 Kink Trajectories in Inhomogeneous DNA
		3.1 Method of Concentrations
		3.2 Method of Blocks and Its Application to Kinks of IFNA17 Gene
		3.3 Kink Trajectories in the pBR322 Plasmid
	4 Conclusions. Further Development and Perspectives of the Methods of Trajectories
	References
Conformational B-A-Transition in the DNA Molecule Model
	References
Soliton Excitations in a Twist-Opening Nonlinear DNA Model
	1 Twist-Opening Nonlinear Model of DNA Double Helix
	2 Dispersion Law
	3 Continuum Approximation
	4 Nonlinear Schrödinger Equation
	5 Korteveg–de Vries Equation
	6 Conclusion
	References
Vibron Self-trapping in Quasi-One-Dimensional Biomolecules: Non-adiabatic Polaron Approach
	1 Introduction
	2 About Energy Processes Inside a Living Cell
		2.1 Hydrolysis of Adenosine Triphosphate
	3 Quasi-1D Biomolecules
		3.1 Proteins: What Is Their Role in the Living Cell?
		3.2 Proteins: What Is Their Basic Structure?
	4 Intra-molecular Vibrational Excitation in Biomolecules: Quasi-Free Excitations or Polarons?
		4.1 The Storage of the Energy Quanta in Biomolecules: Amide-I Mode
		4.2 A Short Excursion to the Absorption Spectra of the Crystalline Acetanilide
		4.3 Beyond Davydov Model
		4.4 Further Investigations in the Framework of Non-adiabatic Polaron Theory
	5 Theory of Exciton Self-trapped States: Non-adiabatic Polaron
		5.1 Starting Hamiltonian
		5.2 Classification of Self-trapped States and Criteria for Their Formation
		5.3 Vibrons in Biomolecules
		5.4 Theory of ST States of a Single Vibron Excitation in Quasi-1D Crystal Structure: Method of the Unitary Transformation
	6 Results and the Discussion
	7 Conclusion
	8 Appendix
		8.1 The Two Useful Relations
		8.2 Some Important Operator Identities
		8.3 Formulas of Lang–Firsov Unitary Transformation
		8.4 The Mean Values of the Functions of Bose Operators
	References
Quantum Correlation Effects in Biopolymer Structures
	1 Introduction
	2 Description of Quantum Correlations
		2.1 Quantum Correlation Functions
		2.2 Properties of the Quantum Correlation Functions
		2.3 Quasi-Distribution Functions and Quantum Characteristic Functions
		2.4 Non-classical Phenomena
		2.5 Entanglement
	3 Model Description of Quantum Correlations in Biomolecules
		3.1 Quantum Mechanical Model of Protein Molecules
		3.2 Influence of the Environment
		3.3 Vibron Quantum Correlations
	4 Conclusion
	References
Nonlinear Dynamics of Microtubules
	1 Introduction
	2 Longitudinal Models for MT Dynamics
		2.1 More General Procedure Within Longitudinal Models for MT Dynamics
		2.2 Application of Morse Potential Energy
	3 Angular Models for MT Dynamics
		3.1 A Series Expansion Unknown Function Method Within the -Model for MT Dynamics
		3.2 General Model for MT Dynamics
	Appendix
	References
Calcium Signaling Along Actin Filaments in Stereocilia Controls Hair-Bundle Motility
	1 Calcium Signaling
	2 Polyelectrolyte Character of Actin Filaments
	3 Models of Pulsatile Waves of Ca2+ Ions Along Actin Filaments
		3.1 Electrochemical Model
		3.2 The Model of F-Actin as a Nonlinear Transmission Line
	4 Ca2+-Dependent Myosin-Based Hair-bundle Motility Adaptation
		4.1 The Coupled Dynamics of Adaptation Motors and Transduction Channels of Stereocilia
	5 Discussion and Conclusions
	References
Theoretical Investigation of Interacting Molecular Motors
	1 Introduction
	2 Molecular Motors
	3 Theoretical Approach (TASEP)
		3.1 Boundary Conditions
		3.2 Update Rules
		3.3 Monte Carlo Simulations: Numerical Approach
		3.4 Master Equation
		3.5 Mathematical Framework
		3.6 Mean-Field Approximation
	4 Development of TASEP Models
	5 Theoretical Model: TASEP with Interactions
		5.1 Model Description
	6 Conclusion
	References