1. Basics of thermodynamics & kinetics
PROTEIN PHYSICS
LECTURES 7-8
Basics of thermodynamics & kinetics

2. THERMODYNAMISC
&
STATISTICAL PHYSICS

3. EXPERIMENTAL DEFINITION :
WHAT IS “TEMPERATURE”?
EXPERIMENTAL DEFINITION :
EXPERIMENTAL DEFINITION
= t,oC o

4. WHAT IS “TEMPERATURE”? THEORY Closed system: energy E = const
S ~ ln[M]
CONSIDER: 1 state of “small part” with  & all
states of thermostat with E-. M(E-) = 1 • Mt(E-)
St(E-)  k • ln[Mt(E-)]
St(E-)  St(E) - •(dSt/dE)|E
Mt(E-)  exp[St(E)/k] • exp[-•(dSt/dE)|E/k]

5. ______________________________________________________________
All-system’s states with E have equal probabilities
For “small part’s” state: depends on e:
COMPARE:
Probability1(1) = Mt(E-1) / M(E) =
exp[- 1• (dSt/dE)|E/k]
and
Probability1(1) = exp(-1/kBT) (BOLTZMANN)
One has: (dSt/dE)|E = 1/ T
k = kB
______________________________________________________________
  -kBT, M  M  exp(1)  M  2.72

6. Pj(j) = exp(-j/kBT)/Z(T); j Pj(j)  1
(dSth/dE) = 1/ T
P1(1) ~ exp(-1/kBT)
Pj(j) = exp(-j/kBT)/Z(T); j Pj(j)  1
Z(T) = i exp(-i/kBT) partition function
СТАТИСТИЧЕСКАЯ СУММА

7. stable Along tangent: S-S(E1) = (E-E1)/ T1
i.e., F = E - T1S = const (= F1 = E1 - T1S1)
stable
Unstable (explodes, v → inf.) Unstable (falls) 
unstable 

8. KIN = mv2/2 : does not depend on coordinates
Separation of potential energy
in classic (non-quantum) mechanics:
P() ~ exp(-/kBT) Classic:  = COORD + KIN
KIN = mv2/2 : does not depend on coordinates
Potential energy COORD: depends only on coordinates
P() ~ exp(-COORD/kBT) • exp(-KIN/kBT)
Z(T) = ZCOORD(T)•ZKIN(T)  F(T) = FCOORD(T)+FKIN(T)
========================================================================================================================
Elementary volume: (mv)x = ħ  (x)3 =(ħ/|mv|)3

9. IN THERMAL EQUILIBRIUM:
TCOORD = TKIN = T
We may consider further
only potential energy:
E  ECOORD
M  MCOORD
S(E)  SCOORD(ECOORD )
F(E)  FCOORD , etc.

10. TRANSITIONS:
THERMODYNAMICS

11. “all-or-none” (or 1st order) phase transition
gradual transition
“all-or-none” (or 1st order) phase transition
coexistence
coexistence
& jump-like
transition
E-T*S=0
Transition: |F1|= |-ST| ~ kT*
(E/kT*)(T/T*) ~ 1

12. Second order phase transition
change
Recently observed in proteins;
they are rare

13. Fhelix_n = Const + nf FICE_n = Cn2/3 + nf
LANDAU: Helix-coil transition: Melting:
NOT 1-s order phase transition s order phase transition N N n n
Helix & coil: 1D objects Ice & water: 3D objects
Fhelix_n = Const + nf FICE_n = Cn2/3 + nf
1D interface D interface
Mid-transition: f = 0
Shelix_n ~ ln(N) positions SICE_n ~ ln(N)
N : very large; n ~ N, <<1 (~0.001)
Const << ln(N) 2/3N2/3 >> ln(N)
phases mix phases do not mix
rule of contraries

14. TRANSITIONS:
KINETICS

15. t01 = t0#1 = n# = n  exp(-F#/kBT) n# n  TRANSITION TIME:
Not
“slowly goes”,
but
climbs, falls
and climbs again… n#
#
falls n 
TRANSITION TIME:
t01 = t0#1 =
= # (n/n#) = #  exp(+F#/kBT)

16. TRANSITION RATE = SUM OF RATES (or: the highest rate)
PARALLEL REACTIONS:
TRANSITION RATE = SUM OF RATES (or: the highest rate)
RATE = 1/ TIME
1/TIME = (1/#)  exp(-F1#/kBT) + (1/#)  exp(-F2#/kBT)

17. _ # # t0… finish = t0#1 finish + t0#2 finish + …
CONSECUTIVE REACTIONS:
TRANSITION TIME 
SUM OF TIMES
(or: the highest time)
start
finish
t0… finish = t0#1 finish + t0#2 finish + …
TIME = #  exp(+F1#/kBT) + #  exp(+F2#/kBT) + …

18. _ _ # main TRANSITION TIME IS ESSENTIALLY
“trap”: on
“trap”: out
start
start
finish
finish
TRANSITION TIME IS ESSENTIALLY
EQUAL FOR “TRAPS” AT AND OUT OF PATHWAYS OF CONSECUTIVE REACTIONS:
TRANSITION TIME  SUM OF TIMES
(or: the longest time)

19. DIFFUSION:
KINETICS

20. Mean kinetic energy of a particle: mv2/2 ~ kBT <> = j Pj(j)  j v2 = (vX2)+(vY2)+(vZ2) Maxwell:
in 3D

21. Friction stops a molecule within picoseconds:
m(dv/dt) = -(3D)v [Stokes law]
D – diameter;
m ~ D3  1g/cm3 – mass;
 – viscosity
tkinet  sec  (D/nm)2 in water
During tkinet the molecule moves somewhere by Lkinet ~ v•tkinet
Then it restores its kinetic energy mv2/2 ~ kBT from thermal kicks of other molecules, and moves in another random side
CHARACTERISTIC DIFFUSION TIME: nanoseconds

22. Friction stops a molecule within picoseconds:
tkinet  sec  (D/nm)2 in water
DIFFUSION:
During tkinet the molecule moves somewhere by Lkinet ~ v•tkinet
Then it restores its kinetic energy
mv2/2 ~ kBT from thermal kicks
of other molecules, and moves in
another random side
CHARACTERISTIC DIFFUSION
TIME: nanoseconds
The random walk allows the molecule
to diffuse at distance D (~ its diameter)
within ~(D/L kinet)2 steps, i.e., within
tdifft  tkinet•(D/Lkinet)2  4•10-10 sec  (D/nm)3 in water

25. STATISTICAL MECHANICS
For “small part”: Pj(j) = exp(-j/kBT)/Z(T);
Z(T) = j exp(-j/kBT)
j Pj(j) = 1
E(T) = <> = j j Pj(j)
if all j =  : #STATES = 1/P, i.e.: S(T) = kBln(1/P)
S(T) = kB<ln(#STATES)> = kBj ln[1/Pj(j)]Pj(j)
F(T) = E(T) - TS(T) = -kBT  ln[ Z(T)]
STATISTICAL MECHANICS

26. Thermostat: Tth = dEth/dSth
“Small part”: Pj(j,Tth) ~ exp(-j/kBTth);
E(Tth) = j j Pj(j,Tth)
S(Tth) = kBj ln[1/Pj(j,Tth)]Pj(j,Tth)
Tsmall_part = dE(Tth)/dS(Tth) = Tth
STATISTICAL
MECHANICS

27. Along tangent: S-S(E1) = (E-E1)/ T1
i.e.,
F = E - T1S = const (= F1 = E1 - T1S1)

28. KIN = mv2/2 : does not depend on coordinates
Separation of potential energy
in classic (non-quantum) mechanics:
P() ~ exp(-/kBT) Classic:  = COORD + KIN
KIN = mv2/2 : does not depend on coordinates
Potential energy COORD: depends only on coordinates
P() ~ exp(-COORD/kBT) • exp(-KIN/kBT)
ZKIN(T) = K exp(-K/kBT): don’t depend on coord.
ZCOORD(T) = Cexp(-C/kBT): depends on coord.
Z(T) = ZCOORD(T)•ZKIN(T)  F(T) = FCOORD(T)+FKIN(T)
========================================================================================================================
Elementary volume: (mv)x = ħ  (x)3 =(ħ/|mv|)3

29. <KIN>   at T<0
P(KIN+COORD) ~ exp(-COORD/kBT)•exp(-KIN/kBT)
P(COORD) = exp(-COORD/kBT) / ZCOORD(T)
ZCOORD(T) = Cexp(-C/kBT): depends ONLY
on coordinates
P(KIN) = exp(-KIN/kBT) / ZKIN(T)
ZKIN(T) = K exp(-K/kBT): don’t depend on coord.
T<0: unstable (explodes)
<KIN>   at T<0
due to
P(KIN) ~ exp(-KIN/kBT)

30. “all-or-none” (or first order) phase transition
F(T1)
________________