Main description:

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.

Contents:

1 Brownian Ratchets and Molecular Motors 3 1.1 The force-generation 3 1.1.1 Maximum Driving Force 3 1.1.2 Stall Force 5 1.2 Smoluchowski-Feynman's ratchet as a heat engine 6 1.2.1 Parrondo Criticism 7 1.3 Ratchet Efficiency 7 1.4 Ratchet Coherency 9 1.5 First Passage Time 9 1.6 Power Stroke 102 The Fokker-Planck equation 17 2.1 The Methods 17 2.2 The Fokker-Planck Equation 18 2.3 Discretization of the Fokker-Planck equation 19 2.3.1 Forward Time Central Space (FTCS) Method 19 2.3.2 Crank-Nicholson Method 20 2.3.3 Stability Analysis 21 2.4 Program 2.1, F-P Equation, Matlab code 253 Biased Brownian Motion 27 3.0.1 Parametrization of the Langevin Equation 27 3.0.2 Numerical Simulations 28 3.0.3 Building the Fokker-Plank's matrices 30 3.0.4 Dichotomous Markov noise 33 3.0.5 Fluctuating Potential, or "Flashing" Ratchet 36 3.0.6 Fluctuating Force, or "Rocking" Ratchet 41 3.1 Programs 46 3.1.1 Program 3.1, Euler Equation, Matlab code 46 3.1.2 Program 3.2, F-P Equation, Matlab code 46 3.1.3 Program 3.3, Dichotomous Noise, Matlab code 48 3.1.4 Program 3.4, Flashing Ratchet, Matlab code
3.1.5 Program 3.5, Rocking Ratchet, Matlab code 4 The Smoluchowski model 49 4.1 Diffusion 52
4.2 Chemical kinetics 54 4.2.1 Absolute Reaction Rate Theory 55 4.3 A mechanochemical model 58 4.3.1 Numerical computation of mechanochemical coupling 59 4.4 Program 4.1, Matlab code 625 Rotation of a dipole. 67 5.1 Introduction 67 5.2 Langevin Equation for the rotor 68 5.3 Dipole in a Ratchet Electrical Potential 70 5.4 Program 5.1, Matlab code 806 Ratchet dimer Brownian motor with Hydrodynamic interactions 84 6.1 Introduction 84 6.2 The Model 85 6.3 Hydrodynamic interactions 88 6.4 Brownian dynamics with hydrodynamic interactions 90 6.4.1 Efficiency 91 6.5 Program 6.1, Fortran code 1007 Fluctuations of the proton electromotive force across inner mitochondrial membrane 115 7.1 Introduction 115 7.2 Theory 116 7.3 Fluctuations of th Proton-Electromotive 116 7.4 Parameter Definitions 118 7.5 Calculation of Buffer Equivalent Electrical Capacitance 118 7.6 Calculation of IMM Electrical Resistance Rm 119 7.7 Relaxation Times of the Electrical and Buffer Reservoirs 119 7.8 Program 7.1, Fluctuations of the PMF, Matlab code 1268 Quantum Ratchets 128 8.1 The Quantum Langevin Equation 128 8.1.1 The Correlation Quantum Function 131 8.1.2 The Quantum Overdamped Langevin Equation with colored noise 132 8.1.3 The Quantum Underdamped Langevin Equation 136 8.1.4 The Ranges 138 8.2 Programs
8.2.1 Program 8.1a, Moderate Damping, Matlab code 8.2.2 Program 8.1b, Complete Langevin Equation, Matlab code Appendices 145
A 01 155 A.1 Master Equation 155 A.1.1 Transition Rate 155 A.1.2 Probability Flux 155 A.1.3 Master Equation 155 A.1.4 Poisson's Process 156 A.1.5 Detailed balance 156B 01 158 B.1 Information Flow 158C 01 163 C.1 Endoreversible Thermodynamics 163D 01 167 D.1 First Passage Phenomena 167 D.1.1 Properties of First Passage Time 167 D.1.2 Application to Chemical Kinetics 168A 02 171 A.1 Stochastic Dynamics 171 A.1.1 White Noise and Wiener Process 171 A.1.2 Spectral Intensity 171 A.1.3 Properties of Wiener's Process 172 A.1.4 Stochastic Process Derivative 173 A.1.5 SDE with Aditive Noise 173 A.1.6 SDE with Multiplicative Noise 174 A.1.7 Ito's and Stratonovich's Calculus 174B 02 179 B.1 Stochastic Energetics 179 B.1.1 Sekimoto View 180 B.1.2 Entropy Production 181 B.1.3 Stochastic Energetics - Useful relations 181A 03 184
A.1 Solution of Equation 184 A.2 Damped oscillations 184B 188 B.1 Electrical and mechanical systems analogies 188C 190 C.1 The fluctuation-dissipation theorem 190D 193 D.1 Integral Algorithm for Colored Noise Simulation 193