HHMI%20meeting,%20FOLDING

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PROTEIN PHYSICS LECTURE 8
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THERMODYNAMISC STATISTICAL PHYSICS
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WHAT IS TEMPERATURE? EXPERIMENTAL DEFINITION
EXPERIMENTAL DEFINITION
t,oC 273.16o
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WHAT IS TEMPERATURE?
THEORY Closed system energy E const
S lnM
CONSIDER 1 state of small part with ? all
states of thermostat with E-?. M(E-?) 1
Mth(E-?) St(E-?) k lnMt(E-?) ? St(E) -
?(dSt/dE)E Mt(E-?) expSt(E)/k
exp-?(dSt/dE)E/k
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All-systems states with E have equal
probabilities For small parts state depends
on e COMPARE Probability1(?1) Mt(E-?1) / S
EM(E) expSt(E)/k 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
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(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
?????????????? ?????
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Along tangent S-S(E1) (E-E1)/ T1
i.e., F E - T1S const
( F1 E1 - T1S1)
stable
Unstable (explodes, V ? ?) Unstable
(fells)
? unstable ?
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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)

E
lementary volume ?(mv)?x h ? (?x)3
(h/mv)3
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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.
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TRANSITIONS THERMODYNAMICS
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gradual transition
all-or-none (or first order) phase transition
coexistence
(?T/T)(?E/kT) 1
coexistence jump
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all-or-none (or first order) phase transition
F(T1)
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Second order phase transition
change
Not observed in proteins up to now they are too
small
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TRANSITIONS KINETICS
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n n ? exp(-?F/kBT)
n
n
?
TRANSITION TIME t0?1 t0?1? ??
(n/n) ?? ? exp(?F/kBT)
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PARALLEL REACTIONS TRANSITION RATE SUM OF
RATES
(or ?the highest rate) RATE 1/ TIME
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_
start
finish
CONSECUTIVE REACTIONS TRANSITION TIME ? SUM OF
TIMES t0? ?finish t0?1? finish t0?2?
finish
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main
main
_
_
trap on
trap out
start
finish
finish
start
TRANSITION TIME IS ESSENTIALLY EQUAL FOR TRAPS
AT AND OUT OF PATHWAYS OF CONSECUTIVE
REACTIONS TRANSITION TIME ? SUM OF TIMES
(or ?the longest
time)
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DIFFUSION KINETICS
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Mean kinetic energy of a particle ?mv2/2?
kBT lt?gt ?j Pj(?j) ? ?j v2
(vX2)(vY2)(vZ2)Maxwell
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• Friction stops a molecule within picoseconds
• m(dv/dt) -(3?D?)v Stokes law
• D diameter
• m D3 mass
• ? viscosity
• tkinet ? 10-13 sec ? (D/nm)2 in water
• During tkinet the molecule moves somewhere by
  Lkinet vtkinet
• 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

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• Friction stops a molecule within picoseconds
• tkinet ? 10-13 sec ? (D/nm)2 in water
• During tkinet the molecule moves somewhere by
  Lkinet vtkinet
• 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

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For small part Pj(?j) exp(-?j/kBT)/Z(T)
Z(T) ?j
exp(-?j/kBT) ?j Pj(?j)
1 E(T) lt?gt ?j ?j? Pj(?j) if all ?j ?
STATES 1/P, i.e. S(T) kB?ln(1/P)
S(T) kBltln(STATES)gt kB??j
ln1/Pj(?j)?Pj(?j) F(T) E(T) - TS(T) -kBT ?
ln Z(T) STATISTICAL MECHANICS
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Thermostat Tth dEth/dSth Small part
Pj(?j,Tth) exp(-?j/kBTth)
E(Tth) ?j ?j? Pj(?j,Tth)
S(Tth) kB??j ln1/Pj(?j,Tth)?Pj(?j,Tth)
Tsmall_part dE(Tth)/dS(Tth) Tth
STATISTICAL MECHANICS
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Along tangent S-S(E1) (E-E1)/ T1
i.e.,
F E - T1S const
( F1 E1 - T1S1)
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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) dont
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 h ? (?x)3
(h/mv)3
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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) dont depend
on coord.
Tlt0 unstable (explodes) lt?KINgt ? ? at Tlt0 due
to P(?KIN) exp(-?KIN/kBT)