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Introduction to Statistical Thermodynamics of Soft and Biological Matter Lecture 3

Introduction to Statistical Thermodynamics of Soft and Biological Matter Lecture 3. Statistical thermodynamics III. Kinetic interpretation of the Boltzmann distribution. Barrier crossing. Unfolding of single RNA molecule. Diffusion.

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Introduction to Statistical Thermodynamics of Soft and Biological Matter Lecture 3

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  1. Introduction to Statistical Thermodynamics of Soft and Biological Matter Lecture 3 Statistical thermodynamics III • Kinetic interpretation of the Boltzmann distribution. • Barrier crossing. • Unfolding of single RNA molecule. • Diffusion. • Random walks and conformations of polymer molecules. • Depletion force.

  2. Boltzmann distribution • System with many possible states (M possible states) • (different conformations of protein molecule) • Each statehas probability • Each state has energy Partition function:

  3. Probability distribution for velocities: Maxwell-Boltzmann distribution - velocity of a molecule Gas of N molecules:

  4. How to compute average… If you want to derive the formula yourself… Use the following help:

  5. verify: Example: fluctuations of polymer molecule - energy of polymer molecule Probability distribution: Equipartition theorem:

  6. Example: Two state system Probability of state: Verify!

  7. - activation barrier Reaction rates: Kinetic interpretation of the Boltzmann distribution

  8. - activation barrier Kinetic interpretation of the Boltzmann distribution Detailed balance (at equilibrium): Number of molecules in state 2 and in state 1 Verify!

  9. Unfolding of single RNA molecule J. Liphardt et al., Science 292, 733 (2001) Optical tweezers apparatus:

  10. Extension Open state: Close state (force applied): extension force Two-state system and unfolding of single RNA molecule J. Liphardt et al., Science 292, 733 (2001)

  11. Diffusion Albert Einstein Robert Brown: 1828 Water molecules (0.3 nm): Pollen grain (1000 nm)

  12. N-th step of random walk: (N-1)-th step of random walk: Verify! Universal properties of random walk One-dimensional random walk: L (step-size of random walk) 0 - random number (determines direction of i-th step)

  13. Diffusion coefficient Number of random steps N corresponds to time t: From dimensional analysis:

  14. Diffusion coefficient and dissipation Einstein relation: Friction coefficient: Viscosity Particle size

  15. Diffusion in two and three dimensions One-dimensional (1D) random walk: Two-dimensional (2D) random walk: Three-dimensional (3D) random walk:

  16. Conformations of polymer molecules L – length of elementary segment • Universal properties of random walk describe conformations • of polymer molecules. * Excluded volume effects and interactions may change law!

  17. Surface area: A x More about diffusion… Diffusion equation Flux: c – concentration of particles

  18. verify this is the solution! c(x,t) x Solution of diffusion equation The concentration profile spreads out with time

  19. Free energy of ideal gas: Pressure: Osmotic forces: Concentration difference induces osmotic pressure Protein solution density: Semi-permeable membrane (only solvent can penetrate) Pressure of ideal gas N – number of particles V - volume

  20. Depletion force R

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