1. PROTEIN PHYSICS LECTURE 8

2. THERMODYNAMISC& STATISTICAL PHYSICS

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

4. THEORYClosedsystem:energy E = const CONSIDER: 1 state of “small part” with  & all states of thermostat with E- . M(E-  ) = 1 M th (E-  ) S t (E-  ) = k ln[M t (E-  )]  S t (E) - (dS t /dE)| E M t (E-  ) = exp[S t (E)/k] exp[- (dS t /dE)| E /k] WHAT IS “TEMPERATURE”? S ~ ln[M]

5. All-system’s states with E have equal probabilities For “small part’s” state: depends on   COMPARE: Probability 1 (  1 ) = M t (E-  1 ) /   E’ M(E’) ~ exp[S t (E)/k] exp[-  1 (dS t /dE)| E /k] exp[S t (E)/k] exp[-  1 (dS t /dE)| E /k]and Probability 1 (  1 ) ~ exp(-  1 /k B T) (BOLTZMANN) One has: (dS t /dE)| E = 1/ T k = k B ______________________________________________________________    -k B T, M  M  exp(1)  M  2.72

6. (dS th /dE) = 1/ T P 1 (  1 ) ~ exp(-  1 /k B T) P j (  j ) = exp(-  j /k B T)/Z(T);  j P j (  j )  1 Z(T) =  i exp(-  i /k B T) partition function СТАТИСТИЧЕСКАЯ СУММА СТАТИСТИЧЕСКАЯ СУММА

7. Unstable (explodes, V →  ): Unstable (fells): stable unstable  Along tangent: S-S(E 1 ) = (E-E 1 )/ T 1 i.e., F = E - T 1 S = const (= F 1 = E 1 - T 1 S 1 ) i.e., F = E - T 1 S = const (= F 1 = E 1 - T 1 S 1 )

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

9. IN THERMAL EQUILIBRIUM: T COORD = T KIN = T We may consider further only potential energy: E  E COORD M  M COORD S(E)  S COORD (E COORD ) F(E)  F COORD, etc.

10. TRANSITIONS:THERMODYNAMICS

11. gradual transition “all-or-none” (or first order) phase transition coexistence & jump coexistence (  T/T * )(  E/kT * ) ~ 1

12. “all-or-none” (or first order) phase transition F(T 1 )

13. Second order phase transition change Not observed in proteins up to now: they are too small

14. TRANSITIONS:KINETICS

15. n # = n  exp(-  F # /k B T) n#n#n#n# n  TRANSITION TIME: t 0  1 = t 0  #1  = =  #  (n/n # ) =  #   exp(+  F # /k B T) =  #  (n/n # ) =  #   exp(+  F # /k B T)

16. PARALLEL REACTIONS: TRANSITION RATE = SUM OF RATES (or:  the highest rate) (or:  the highest rate) RATE = 1/ TIME

17. CONSECUTIVE REACTIONS: TRANSITION TIME  SUM OF TIMES t 0  …  finish = t 0  #1  finish + t 0  #2  finish + … # # startfinish _

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

19. DIFFUSION:KINETICS

20. Mean kinetic energy of a particle:  mv 2 /2  ~ k B T =  j P j (  j )   j v 2 = (v X 2 )+(v Y 2 )+(v Z 2 ) Maxwell:

21. Friction stops a molecule within picoseconds: m(dv/dt) = -(3  D  )v [Stokes law] D – diameter; m ~ D 3 – mass;  – viscosity t kinet  10 -13 sec  (D/nm) 2 in water During t kinet the molecule moves somewhere by L kinet ~ vt kinet Then it restores its kinetic energy mv 2 /2 ~ k B T from thermal kicks of other molecules, and moves in another random side CHARACTERISTIC DIFFUSION TIME: nanoseconds

22. Friction stops a molecule within picoseconds: t kinet  10 -13 sec  (D/nm) 2 in water During t kinet the molecule moves somewhere by L kinet ~ vt kinet Then it restores its kinetic energy mv 2 /2 ~ k B T 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 t difft  t kinet (D/L kinet ) 2  410 -10 sec  (D/nm) 3 in water

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

25. Thermostat: T th = dE th /dS th “Small part”: P j (  j,T th ) ~ exp(-  j /k B T th ); E(T th ) =  j  j  P j (  j,T th ) E(T th ) =  j  j  P j (  j,T th ) S(T th ) = k B   j ln[1/P j (  j,T th )]  P j (  j,T th ) S(T th ) = k B   j ln[1/P j (  j,T th )]  P j (  j,T th ) T small_part = dE(T th )/dS(T th ) = T th STATISTICALMECHANICS

26. Along tangent: S-S(E 1 ) = (E-E 1 )/ T 1 i.e., i.e., F = E - T 1 S = const (= F 1 = E 1 - T 1 S 1 ) F = E - T 1 S = const (= F 1 = E 1 - T 1 S 1 )

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

28. P(  KIN +  COORD ) ~ exp(-  COORD /k B T)exp(-  KIN /k B T) P(  COORD ) = exp(-  COORD /k B T) / Z COORD (T) Z COORD (T) =  C exp (-  C /k B T): depends ONLY on coordinates on coordinates P(  KIN ) = exp(-  KIN /k B T) / Z KIN (T) Z KIN (T) =  K exp (-  K /k B T): don’t depend on coord. T<0: unstable (explodes)   at T   at T<0 due to P(  KIN ) ~ exp(-  KIN /k B T)