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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

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

More about diffusion… Diffusion equation

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



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