Introduction to Statistical Thermodynamics of Soft and Biological Matter Lecture 2 Statistical thermodynamics II Free energy of small systems. Boltzmann.

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Presentation transcript:

Introduction to Statistical Thermodynamics of Soft and Biological Matter Lecture 2 Statistical thermodynamics II Free energy of small systems. Boltzmann distribution. Partition function. Probability distributions. Fluctuations. Free energy of two-state system. Kinetic interpretation of the Boltzmann distribution. Barrier crossing. Unfolding of single RNA molecule.

Some more soft and biological matter… Polymers Gels Polymers – linear macromolecules Amphiphiles: soaps, lipids, membranes polar head (love water) hydrocarbon tail (hate water) Self-assembly

And more biological matter… Actin in cell Cell on substrate bilayer vesicle Vesicle shapes

Entropy… Probability of each state: Molecules A Molecules B

Hard-sphere crystal Hard-sphere liquid Hard-sphere freezing is driven by entropy ! Lower Entropy… Higher Entropy…

Entropy of ideal gas Indistinguishablility For N molecules: For one molecule: - “cell” volume (quantum uncertainty ) V – total volume Free energy of ideal gas: density:

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

Free energy Reservoir, T Small system For closed system: a For open (small) system Free energy is minimized: - Helmholtz free energy - Gibbs free energy

Protein molecule with several possible conformations Free energy Reaction coordinate (order parameter) M Free energy landscape:

Boltzmann distribution System with many possible states ( M possible states) (different conformations of protein molecule) Each state has probability Each state has energy - normalization condition Free energy (per one protein molecule): Minimize free energy with respect to : Constraint minimization

Minimize free energy: Subject to constraint: Method of Lagrange Multipliers (look at any book on calculus): Lagrange multiplier

Partition function Free energy: Minimize free energy. Solution: Substitute into F: Partition function:

Sequence Analysis Course… Lecture 9 Boltzmann equation

Self-Consistent Mean Field (SCMF) modeling DNA/Protein structure-function analysis and prediction. Lecture 10 …

The notion of probability distribution Probability distribution function: - random variable - probability to find in the interval - normalization condition - average value of Fluctuation (variance):

Examples of probability distributions Gaussian probability distribution: Your turn: find A variance: Uniform probability distribution:

Example: fluctuations of spring in thermal bath verify: - energy of spring - probability distribution Equipartition theorem:

Example: Two state system

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

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

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