Lecture 4 Barometric formula and the Boltzmann equation

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

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

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

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 DE 3 -2 the fraction of particles in each state: 1 - partition function DE 2 -1

The energy difference here represents enthalpy H = U + W (internal energy +work) pi pj S = k ln. W W is the number of micro-states DH pi/pj e-1 = 0. 37 e-2 = 0. 135 e-3 = 0. 05 e-4 = 0. 018 e-5 = 0. 007 DH Free energy difference D G = D H - TD S 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: omp. F (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

First Met is usually cleaved off Primary sequence of human myoglobin 1 41 81 121 mglsdgewql lekfdkfkhl ghheaeikpl pgdfgadaqg vlnvwgkvea ksedemkase aqshatkhki amnkalelfr N-terminus (amino group) dipghgqevl dlkkhgatvl pvkylefise kdmasnykel irlfkghpet talggilkkk ciiqvlqskh gfqg C-terminus (carboxyl group) …go to protein database

Myoglobin family tree

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