1. Lecture 4 Barometric formula and the Boltzmann equation
Simple notions on Free Energy
Proteins
Reading: Chapter 3

2. h Barometric formula (let’s quickly derive…) h
a particle above the ground h
a column of uniform fluid h
pressure at the bottom
potential energy =
mass × gravitational const. × height = work against the gravity
pressure = density × gravitational const. × height

3. T is constant, but density depends on height
Now we have a column of compressible gas in the gravity field:
T is constant, but density depends on height
density
ideal gas equation dh
n – number of particles
m - mass
substituting:
Barometric formula po

4. c = concentration (which is probability)
Barometric formula
because pressure is
proportional to the number of particles p ~ n
n = number of particles
per unit volume
normalizing to the volume c = n/V
c = concentration (which is probability)
in our case U is constant because T is constant

5. Boltzmann equation uses probabilities
the relative populations of particles in states i and j separated by an energy gap 3 2
DE3-2
the fraction of particles in each state:
DE2-1 1
- partition function

6. S = k lnW pi pj pi/pj The energy difference here represents enthalpy
H = U + W (internal energy +work)
W is the number of micro-states pi pj DH DH
Free energy difference DG = DH - TDS
pi/pj
e-1 = 0.37
e-2 = 0.135
e-3 = 0.05
e-4 = 0.018
e-5 = 0.007
entropic advantage
For two global states which can be ensembles of microstates:

7. Proteins
Expression of genetic information: blueprint to structure/machine
Should have emergent properties…catalytic, binding, motor, control, transport, …

9. Hierarchy
Folding order

13. Alpha helix
Beta sheet

14. Beta barrel channel: ompF (E. coli)

15. Dependent on the size and flexibility of side chains, only limited ranges of Phi (Φ) and Psy (Ψ) angles are permitted

18. Residues forming hairpins are not in helical or b-sheet conformations
Glycines frequently occur in turns and loops because they can occupy essentially the entire Phi-Psy space

19. Different representations of structures (PDB coordinates)

20. coiled coils are predicted by amphipathic character of helixes and heptad (7-residue repeats) organization

22. Primary sequence of human myoglobin
First Met is usually cleaved off
Primary sequence of human myoglobin
mglsdgewql vlnvwgkvea dipghgqevl irlfkghpet
41 lekfdkfkhl ksedemkase dlkkhgatvl talggilkkk
81 ghheaeikpl aqshatkhki pvkylefise ciiqvlqskh
121 pgdfgadaqg amnkalelfr kdmasnykel gfqg
N-terminus (amino group) C-terminus (carboxyl group)
…go to protein database

23. Myoglobin family tree

24. The 4-subunit association of Hb confers cooperativity of oxygen binding