Introduction to Statistical Thermodynamics of Soft and Biological Matter Lecture 3

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# Introduction to Statistical Thermodynamics of Soft and Biological Matter Lecture 3 - PowerPoint PPT Presentation

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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Presentation Transcript

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.

Boltzmann distribution

• System with many possible states (M possible states)
• (different conformations of protein molecule)
• Each statehas probability
• Each state has energy

Partition function:

Probability distribution for velocities:

Maxwell-Boltzmann distribution

- velocity of a molecule

Gas of N molecules:

How to compute average…

If you want to derive the formula yourself…

Use the following help:

verify:

Example: fluctuations of polymer molecule

- energy of polymer molecule

Probability distribution:

Equipartition theorem:

Example: Two state system

Probability of state:

Verify!

- activation barrier

Reaction rates:

Kinetic interpretation of the

Boltzmann distribution

- activation barrier

Kinetic interpretation of the

Boltzmann distribution

Detailed balance (at equilibrium):

Number of molecules in state 2 and in state 1

Verify!

Unfolding of single RNA molecule

J. Liphardt et al., Science 292, 733 (2001)

Optical tweezers apparatus:

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)

Diffusion

Albert Einstein

Robert Brown: 1828

Water molecules (0.3 nm):

Pollen grain (1000 nm)

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)

Diffusion coefficient

Number of random steps N corresponds to time t:

From dimensional analysis:

Diffusion coefficient and dissipation

Einstein relation:

Friction coefficient:

Viscosity

Particle size

Diffusion in two and three dimensions

One-dimensional (1D) random walk:

Two-dimensional (2D) random walk:

Three-dimensional (3D) random walk:

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!

Surface area: A

x

Flux:

c – concentration of particles

verify this is the solution!

c(x,t)

x

Solution of diffusion equation

The concentration profile spreads out with time

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