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

2. THERMODYNAMISC & STATISTICAL PHYSICS

3. WHAT IS “TEMPERATURE”? EXPERIMENTAL DEFINITION : EXPERIMENTAL DEFINITION = t,oC + 273.16o

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. (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. 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. 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. 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. LANDAU: Helix-coil transition: Melting: NOT 1-s order phase transition 1-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 3D 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. n#=nexp(-F#/kBT) Not “slowly goes”, but climbs, falls and climbs again… # n# falls n  TRANSITION TIME: t01 = t0#1 = = # (n/n#)=#exp(+F#/kBT)

16. 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. # # _ 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 main# _ _ “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 10-13 sec  (D/nm)2in 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 10-13 sec  (D/nm)2in 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)2steps, i.e., within tdifft tkinet•(D/Lkinet)2 4•10-10 sec  (D/nm)3in water

25. 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. 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. 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) ________________