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
sol
gel
Polymers – linear macromolecules
• Amphiphiles: soaps, lipids, membranes
• polar head (love water)
• hydrocarbon tail (hate water)
Self-assembly
And more biological matter…
Actin in cell
bilayer vesicle
Cell on substrate
Vesicle shapes
Entropy…
Molecules A
Molecules B
Probability of each state:
Hard-sphere liquid
Lower Entropy…
Hard-sphere freezing is driven
by entropy !
Higher Entropy…
Hard-sphere crystal
Entropy of ideal gas
For one molecule:
V – total volume
- “cell” volume (quantum uncertainty )
For N molecules:
Indistinguishablility
Free energy of ideal gas:
density:
Pressure of ideal gas
Free energy of ideal gas:
N – number of particles
V - volume
density:
Pressure:
Osmotic forces:
Protein solution
Concentration difference induces
osmotic pressure
Semi-permeable membrane
(only solvent can penetrate)
Free energy
For closed system:
a
Small system
For open (small) system Free energy is minimized:
Reservoir, T
- Helmholtz free energy
- Gibbs free energy
Protein molecule with several
possible conformations
Free energy
Free energy landscape:
2
M
1
3
Reaction coordinate (order parameter)
Boltzmann distribution
• System with many possible states (M possible states)
(different conformations of protein molecule)
Each state has probability
Each state has energy
Free energy (per one protein molecule):
- normalization condition
Constraint minimization
• 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
Partition function:
into F:
Sequence Analysis Course… Lecture 9
Boltzmann equation
DNA/Protein structure-function analysis
and prediction. Lecture 10…
Self-Consistent Mean Field (SCMF) modeling
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:
2
Uniform probability distribution:
1.75
1.5
1.25
1
0.75
0.5
0.25
0.25
0.5
0.75
1
1.25
1.5
1.75
2
Example: fluctuations of spring
in thermal bath
- energy of spring
verify:
- 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:
Two-state system and unfolding
of single RNA molecule
J. Liphardt et al., Science 292, 733 (2001)
Extension
Open state:
Close state (force applied):
force
extension