Lecture 4 Barometric formula and the Boltzmann equation

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Presentation transcript:

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

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

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

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

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

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:

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

Hierarchy Folding order

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 b-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

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

Myoglobin family tree

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