Statistical mechanics, the partition function, and fj rst- order - PowerPoint PPT Presentation

Protein Physics 2016 Lecture 5, February 2 Statistical mechanics, the partition function, and fj rst- order phase transitions Magnus Andersson magnus.andersson@scilifelab.se Theoretical & Computational Biophysics

Recap • Secondary structure & turns • Properties, simple stability concepts • Geometry/topology • Amino acid properties, titration • Natural selection of residues in proteins • Free energy of hydrogen bond formation in proteins when in vacuo or aqueous solvent

titratable amino acids

F = E-TS F = E-T(k lnV) i.e. number of accessible states probability ∝ exp(-F/kT) E H <0: Enthalpy of a 
 hydrogen bond S water >0: Entropy of freely rotating body or complex (1 or 2 waters!)

Recap: H-bond Δ G State A State B In vacuo D A D Δ G? A E b =E H ,S b =0 E a =0,S a =0 In solvent D D A A Δ G? E b =2E H ,S b =S H E a =2E H ,S a =0

Today • Statistical mechanics • The partition function • Free energy & stable states • Gradual changes & phase transitions • Activation barriers & transition kinetics

Fluctuations in a closed system - E conserved Consider all microstates of this system with energy E # thermostat microstates M therm with the energy (E- ɛ ) De fj ne: S = k*ln M therm

Entropy ⇥ ⇤ S therm( E − ✏ ) =  ln M therm( E − ✏ ) Now do series expansion; only 1st order matters - why? ◆� ✓ dS therm � S therm( E − ✏ ) = S therm( E ) − ✏ � dE � E Solve for M  S therm( E − ✏ ) � M ( E − ✏ ) = exp   S therm( E )  ( dS therm /dE ) | E � ⇢ �� = exp × exp − ✏  

Observation of microstates • The probability of observing the small part in this state is proportional to the number of microstates corresponding to it p ∝ M ( E − ✏ ) ∝ exp { − ✏ [( dS/dE ) | E /  ] } ( dS/dE ) | E = 1 κ = k T

Energy increases What happens when energy increases by kT? ln [ M ( E + k B T )] = S ( E + k B T ) /k B = = [ S ( E ) + k B (1 /T )] /k B = ln [ M ( E )] + 1 e (2.72) times more microstates, 
 regardless of system properties and size!

Probabilities of states probability of being in a state i w i ( T ) = exp ( − � i /k B T ) Z ( T ) Normalization factor � Z ( T ) = exp ( − � i /k B T ) i ‘The partition function’

X E ( T ) = w i ✏ i i X S ( T ) = w i S i i How do we calculate S i ?

System distribution over states Consider N systems - how can we distribute them? w 1 w 2 w 3 w j 1 2 3 j X n i = w i N w i N = N i Question: how many ways can these systems be distributed over the j states?

Permutations N ! n 1 ! n 2 ! ...n j ! = Stirling: n! ≈ (n/e) n = ( N/n 1 ) n 1 ... ( N/n j ) n j = (1 /w 1 ) Nw 1 ... (1 /w j ) Nw j ⇤ N 1 ...w w j 1 / ( w w 1 ⇥ = j ) for N systems 1 ...w w j = 1 / ( w w 1 j ) for 1 system ...and the entropy becomes: X S = k B ln M = k B w i ln(1 /w i ) i

Free energy X E ( T ) = w i ✏ i i X S = k B ln M = k B w i ln(1 /w i ) i F = E - TS F = -kBT ln [Z(T)]

System instability

System stability

Gradual changes What does this correspond to? Examples?

Abrupt changes dT = 4kT^2 / E2-E1 A first-order phase transition!

A di ff erent change... A second-order phase transition!

Free energy barriers n # ≈ n exp(- ∆ F # /k B T) Ƭ (n/n # ) ≈ Ƭ exp( ∆ F # /k B T) + ∆ F # /k B T � ⇥ t 0 → 1 ≈ τ exp k 0 → 1 = 1 /t 0 → 1 Transition rate:

Secondary structure • Alpha helix formation • Equilibrium between helix & coil • Beta sheet formation • Properties of the “random” coil, 
 or the denatured state - what is it?

Alpha helix formation • Hydrogen bonds: i to i+4 • 0-4, 1-5, 2-6 • First hydrogen bond “locks” 
 residues 1,2,3 in place • Second stabilizes 2,3,4 (etc.) • N residues stabilized by N-2 hydrogen bonds!

Alpha helix free energy • Free energy of helix vs. “coil” states: Entropy loss of fj xating one H-bond free energy number of residues residue in helix ∆ F α = F α − F coil = ( n − 2) f H-bond − nTS α � ⇥ = − 2 f H-bond + n f H-bond − TS α Helix initiation Helix cost elongation cost ∆ F α = f INIT + nf EL

Alpha helix free energy � ⇥ � ⇥ exp ( − ∆ F α /k B T ) = exp − f INIT /k B T exp − nf EL /k B T ⇥⌅ n � ⇥ ⇤ � = exp − f INIT /k B T exp − f EL /k B T σ s n = � ⇥ s − f EL /k B T = exp � ⇥ − f INIT /k B T = exp σ � ⇥ � ⇥ − f INIT /k B T +2 f H /k B T << 1 = exp = exp σ Equilibrium constant for helix of length n

How does a helix form? • First, consider ice in water n ∝ V ∝ r 3 A ∝ r 2 ∝ n 2 / 3 Surface tension costly! • S = k ln(N)

How does a helix form? • Landau: Phases cannot co-exist in 3D • First order phase transitions means either state can be stable, but not the mixture • Think ice/water - either freezing or melting 
 n ∝ V ∝ r 3 A ∝ r 2 ∝ n 2 / 3 Surface tension costly! • But a helix-coil transition in a chain is 1D! • Interface helix/coil does not depend on n 


How does a helix form? ice/water: n molecules in ice, N in water energy cost * n^2/3 & entropy: k ln N helix/coil: n residues in helix out of N in total f INIT - kT ln (N-n) i.e. opposite to water/ice!


 Helix/coil mixing • Or: What helix length corresponds to the 
 transition mid-point? f EL = f H − TS α = 0 • Assuming helix can start/end anywhere, 
 there are N^2/2 positions 
 S = k ln V ≈ k ln N 2 = 2 k ln N ∆ F helix ≈ f INIT − 2 kT ln N • At transition midpoint we have Δ F=0 & N=n 0 
 � ⇥ = 1 / √ σ n 0 = exp f INIT / 2 kT

Helix parameters • We can measure n 0 from CD-spectra • Calculate σ from last equation • Typical values for common amino acids: 
 n 0 ≈ 30 f INIT ≈ 4 kcal/mol σ ≈ 0.001 • f H = -f INIT /2 = -2 kcal/mol • TS α = f H - f EL ≈ -2 kcal/mol 
 (Conformational entropy loss of helix res.)

Helix stability • Temperature dependence • Elongation term dominant for large n 0 • dF(alpha) = f INIT + n 0 *f EL Highly cooperative, but NOT a formal 
 phase transition! (width does 
 not go to zero)

Helix studies • CD spectra • Determine s & σ • Alanine: s ≈ 2, f EL ≈ -0.4kcal/mol • Glycine: s ≈ 0.2, f EL ≈ +1kcal/mol • Proline: s ≈ 0.01-0.001 , f EL ≈ +3-5kcal/mol • Bioinformatics much more e ffi cient for prediction, though!

Rate of Formation • Experimentally: Helices form in ~0.1 μ s! 
 (20-30 residue segments) • One residue < 5 ns... What is the 
 limiting step?


 Formation... τ :1-residue 
 • Rate of formation at position 1: 
 elongation � ⇥ t INIT0 = τ exp f INIT /kT = τ / σ • Rate of formation anywhere (n0 ≈ 1/ √σ ): 
 t INIT = τ / √ σ • Propagation to all residues: tn 0 = τ / √ σ • Half time spent on initiation, half elongation!

Helix summary • Very fast formation • Both initiation & elongation matters • Quantitative values derived from CD-spectra • Low free energy barriers, ~1kcal/mol • Characteristic lengths 20-30 residues