Download - 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
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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
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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
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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
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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
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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 /
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Proteins
•Expression of genetic information: blueprint to structure/machine
•Should have emergent properties…catalytic, binding, motor, control, transport, …
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Folding order Hierarchy
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Alpha helix Beta sheet
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Beta barrel channel: ompF (E. coli)
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Dependent on the size and flexibility of side chains, only limited ranges of Phi (Φ) and Psy (Ψ) angles are permitted
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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
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Different representations of structures (PDB coordinates)
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coiled coils are predicted by amphipathic character of helixes and heptad (7-residue repeats) organization
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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
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Myoglobin family tree
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The 4-subunit association of Hb confers cooperativity of oxygen binding