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