lecture 4 barometric formula and the boltzmann equation simple notions on free energy proteins
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Lecture 4
Barometric formula and the Boltzmann equation
Simple notions on Free Energy
Proteins
Reading: Chapter 3
a column of uniform fluid
ghp
pressure at the bottom
Barometric formula (let’s quickly derive…)
pressure = density × gravitational const. × height
ghmEpot h
a particle above the ground
potential energy = mass × gravitational const. × height = work against the gravity
h
Now we have a column of compressible gas in the gravity field:
T is constant, but density depends on height
gdhhdp )(
dh
V
nm nkTpV
ideal gas equation
n – number of particlesm - mass
density
pdhkT
mgdp substituting:
hp
po
dhkT
mg
p
dp
0
kT
mgh
p
p
o
lnkTmgh
oepp /
dpp
p
Barometric formula
po
kTmgh
o
ep
p /Barometric formula
kTE
o
potec
c /
kTmgh
o
en
n / n = number of particles per unit volume
c = concentration (which is probability)
because pressure is proportional to the number of
particles p ~ n
normalizing to the volume c = n/V
potEUH kTH
o
ec
c /
in our case U is constant because T is constant
Boltzmann equation uses probabilities
kTEE
j
i jiep
p /)(
the relative populations of particles in states i and j separated by an energy gap
t
j
kTE
kTE
jj
j
e
ep
1
/
/
t
j
kTE je1
/- partition function
the fraction of particles in each state:
E2-1
E3-2
1
3
2
S = k lnW
Free energy difference G = H - TS
W is the number of micro-states
e-1 = 0.37
e-2 = 0.135
e-3 = 0.05
e-4 = 0.018
e-5 = 0.007
H
entropic advantage
The energy difference here represents enthalpyH = U + W (internal energy +work)
kTG
j
i ep
p /
For two global states which can be ensembles of microstates:
H
pi/pj
pi
pj
kTH
j
i ep
p /
Proteins
•Expression of genetic information: blueprint to structure/machine
•Should have emergent properties…catalytic, binding, motor, control, transport, …
Folding order Hierarchy
Alpha helix Beta sheet
Beta barrel channel: ompF (E. coli)
Dependent on the size and flexibility of side chains, only limited ranges of Phi (Φ) and Psy (Ψ) angles are permitted
Residues forming hairpins are not in helical or -sheet conformations
Glycines frequently occur in turns and loops because they can occupy essentially the entire Phi-Psy space
Different representations of structures (PDB coordinates)
coiled coils are predicted by amphipathic character of helixes and heptad (7-residue repeats) organization
1 mglsdgewql vlnvwgkvea dipghgqevl irlfkghpet41 lekfdkfkhl ksedemkase dlkkhgatvl talggilkkk81 ghheaeikpl aqshatkhki pvkylefise ciiqvlqskh 121 pgdfgadaqg amnkalelfr kdmasnykel gfqg
Primary sequence of human myoglobin
N-terminus (amino group) C-terminus (carboxyl group)
First Met is usually cleaved off
…go to protein database
Myoglobin family tree
The 4-subunit association of Hb confers cooperativity of oxygen binding
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