Principles of Brownian and Molecular Motors (Springer Series in Biophysics, 21)

Molecular motors convert chemical energy (typically from ATP hydrolysis) to directed motion and mechanical work. Biomolecular motors are proteins able of converting chemical energy into mechanical motion and force. Because of their dimension, the many small parts that make up molecular motors must operate at energies only a few times greater than those of the thermal baths. The description of molecular motors must be stochastic in nature. Their actions are often described in terms of Brownian Ratchets mechanisms. In order to describe the principles used in their movement, we need to use the tools that theoretical physics give us. In this book we centralize on the some physical mechanisms of molecular motors. Preface Contents 1 Brownian Ratchets and Molecular Motors 1.1 The Force-Generation 1.1.1 Maximum Driving Force 1.1.2 Stall Force 1.2 Smoluchowski-Feynman's Ratchet as a Heat Engine 1.2.1 Parrondo Criticism 1.3 Ratchet Efficiency 1.4 Ratchet Coherency 1.5 First Passage Time 1.6 Power Stroke Bibliography 2 The Fokker-Planck Equation 2.1 The Methods 2.2 The Fokker-Planck Equation 2.3 Discretization of the Fokker-Planck Equation 2.3.1 Forward Time Central Space (FTCS) Method 2.3.2 Crank-Nicholson Method 2.3.3 Stability Analysis 2.4 Program 2.1, F-P Equation, Matlab Code Bibliography 3 Biased Brownian Motion 3.1 Parametrization of the Langevin Equation 3.2 Numerical Simulations 3.3 Building the Fokker-Plank's Matrices 3.3.1 Periodic Potential Slightly Tilted 3.4 Dichotomous Markov Noise 3.4.1 Generating Dichotomous Markov Noise 3.5 Fluctuating Potential, or ``Flashing'' Ratchet 3.6 Fluctuating Force, or ``Rocking'' Ratchet 3.7 Programs 3.7.1 Program 3.1, Euler Equation, Matlab Code 3.7.2 Program 3.2, F-P Equation, Matlab Code 3.7.3 Program 3.3, Dichotomous Noise, Matlab Code 3.7.4 Program 3.4, Flashing Ratchet, Matlab Code 3.7.5 Program 3.5, Rocking Ratchet, Matlab Code Bibliography 4 The Smoluchowski Model 4.1 Diffusion 4.2 Chemical Kinetics 4.2.1 Absolute Reaction Rate Theory 4.3 A Mechanochemical Model 4.3.1 Numerical Computation of Mechanochemical Coupling 4.4 Program 4.1, Matlab Code Bibliography 5 Rotation of a Dipole 5.1 Introduction 5.2 Langevin Equation for the Rotor 5.3 Dipole in a Ratchet Electrical Potential 5.4 Program 5.1, Matlab Code Bibliography 6 Ratchet Dimer Brownian Motor with Hydrodynamic Interactions 6.1 Introduction 6.2 The Model 6.3 Hydrodynamic Interactions 6.4 Brownian Dynamics with Hydrodynamic Interactions 6.4.1 Efficiency 6.5 Program 6.1, Fortran Code Bibliography 7 Fluctuations of the Proton Electromotive Force Across Inner Mitochondrial Membrane 7.1 Introduction 7.2 Theory 7.3 Fluctuations of the Proton-Electromotive 7.4 Parameter Definitions 7.5 Calculation of Buffer Equivalent Electrical Capacitance 7.6 Calculation of IMM Electrical Resistance Rm 7.7 Relaxation Times of the Electrical and Buffer Reservoirs 7.8 Program 7.1, Fluctuations of the PMF, Matlab Code Bibliography 8 Quantum Ratchets 8.1 The Quantum Langevin Equation 8.1.1 The Correlation Quantum Function 8.1.1.1 Dimensionless Parameters 8.1.2 The Quantum Overdamped Langevin Equation with Colored Noise 8.1.2.1 The Dimensionless Quantum Overdamped Langevin Equation 8.1.3 The Quantum Underdamped Langevin Equation 8.1.4 The Ranges 8.1.4.1 Classical Range 8.1.4.2 Classical Smoluchowski Range 8.1.4.3 Quantum Range 8.1.4.4 Quantum Smoluchowski Range 8.1.4.5 Discretization of the Quantum Underdamped Langevin Equation 8.2 Programs 8.2.1 Program 8.1a, Moderate Damping, Matlab Code 8.2.2 Program 8.1b, Complete Langevin Equation, MatlabCode Bibliography Appendix A A.1 Master Equation A.1.1 Transition Rate A.1.2 Probability Flux A.1.3 Master Equation A.1.4 Poisson's Process A.1.5 Detailed Balance Appendix B B.1 Information Flow Bibliography Appendix C C.1 Endoreversible Thermodynamics Bibliography Appendix D D.1 First Passage Phenomena D.1.1 Properties of First Passage Time D.1.2 Application to Chemical Kinetics Bibliography Appendix E E.1 Stochastic Dynamics E.1.1 White Noise and Wiener Process E.1.2 Spectral Intensity E.1.3 Properties of Wiener's Process E.1.4 Stochastic Process Derivative E.1.5 SDE with Aditive Noise E.1.6 SDE with Multiplicative Noise E.1.7 Itô's and Stratonovich's Calculus E.1.7.1 Convergence in Quadratic Mean E.1.7.2 Properties of Itô's Integral E.1.7.3 Itô's Differential E.1.7.4 Itô's Stochastic Differential Equation E.1.7.5 Stratonovich's Calculus E.1.7.6 Stratonovich's Stochastic Differential Equation Bibliography Appendix F F.1 Stochastic Energetics F.1.1 Sekimoto View F.1.2 Entropy Production F.1.3 Stochastic Energetics: Useful Relations F.1.3.1 Jarzynski's Equality F.1.3.2 Crooks Theorem Bibliography Appendix G G.1 Solution of Equation G.2 Damped Oscillations G.2.1 Clasification G.2.2 Summary G.2.2.1 (a) Overdamped Oscillations, γ2 ω0 G.2.2.2 (b) Underdamped Oscillations, γ2 < ω0 G.2.2.3 (c) Critical Damping Oscillations, γ2 = ω0 Appendix H H.1 Electrical and Mechanical Systems Analogies Appendix I I.1 The Fluctuation-Dissipation Theorem Bibliography Appendix J J.1 Integral Algorithm for Colored Noise Simulation Bibliography