PROTEIN PHYSICSPROTEIN PHYSICS

LECTURE 8LECTURE 8

THERMODYNAMISCTHERMODYNAMISC&&

STATISTICAL PHYSICSSTATISTICAL PHYSICS

WHAT IS “TEMPERATURE”?WHAT IS “TEMPERATURE”?

EXPERIMENTAL DEFINITION :EXPERIMENTAL DEFINITION :

= t,oC + 273.16o

EXPERIMENTAL DEFINITION

THEORYTHEORY

ClosedClosedsystem:system:energy energy E = constE = const

CONSIDER: 1 state of “small part” with CONSIDER: 1 state of “small part” with & all & all states of thermostat with E-states of thermostat with E-.. M(E- M(E-) = 1) = 1 •• MMthth(E-(E-) )

SStt(E-(E-) = ) = k k •• ln[Mln[Mtt(E-(E-)] )] S Stt(E) - (E) - ••(dS(dStt/dE)|/dE)|E E

MMtt(E-(E-) = exp[S) = exp[Stt(E)/(E)/kk] ] • • exp[-exp[-••(dS(dStt/dE)|/dE)|EE//kk]]

WHAT IS “TEMPERATURE”?WHAT IS “TEMPERATURE”?

S ~S ~ ln[M]ln[M]

All-system’s states with E have All-system’s states with E have equalequal probabilitiesprobabilitiesFor “small part’s” state: For “small part’s” state: depends on depends on COMPARE:COMPARE:

ProbabilityProbability11((11) = M) = Mtt(E-(E-11)) // E’E’M(E’)M(E’) ~ ~

exp[Sexp[Stt(E)/(E)/kk] ] • • exp[-exp[- 11• • (dS(dStt/dE)|/dE)|EE//kk]]andand

ProbabilityProbability11((11) ~ exp(-) ~ exp(-11/k/kBBT) T) (BOLTZMANN)(BOLTZMANN)

One has: One has: (dS(dStt/dE)|/dE)|EE = 1/ = 1/ T T

k = k = kkBB____________________________________________________________________________________________________________________________

-k-kBBT, T, M M M M exp(1) exp(1) M M 2.72 2.72

(dS(dSthth/dE) = 1//dE) = 1/ TT

PP11((11) ~ exp(-) ~ exp(-11/k/kBBT)T)

PPjj((jj) = exp(-) = exp(-jj/k/kBBT)/Z(T); T)/Z(T); j j PPjj((jj) ) 1 1

Z(T) = Z(T) = i i exp(-exp(-ii/k/kBBT) T) partition functionpartition function СТАТИСТИЧЕСКАЯ СУММАСТАТИСТИЧЕСКАЯ СУММА

Unstable (explodes, V Unstable (explodes, V → → ): Unstable (fells):): Unstable (fells):

stablestable

unstableunstable

Along tangent: S-S(EAlong tangent: S-S(E11)) = (E-E= (E-E11)/)/ TT11

i.e.,i.e., F = E - TF = E - T11SS = = constconst (= F (= F11 = E = E11 - T - T11SS11) )

Separation of potential energySeparation of potential energyin classic (non-quantum) mechanics:in classic (non-quantum) mechanics:

P(P() ~ exp(-) ~ exp(-/k/kBBT) T) Classic:Classic: == COORDCOORD + + KINKIN

KINKIN = mv= mv22/2 : /2 : does not depend on coordinatesdoes not depend on coordinates

Potential energyPotential energy COORDCOORD: : depends only on coordinatesdepends only on coordinates

P(P() ~ exp(-) ~ exp(-COORDCOORD/k/kBBT) T) • • exp(-exp(-KINKIN/k/kBBT)T)

Z(T) = ZZ(T) = ZCOORDCOORD(T)(T)••ZZKINKIN(T) (T) F(T) = F F(T) = FCOORDCOORD(T)(T)+F+FKINKIN(T) (T) ================================================================================================================================================================================================================================================

Elementary volume: Elementary volume: (mv)(mv)x = x = ħ ħ ( (x)x)33 =( =(ħ/|mv|)ħ/|mv|)33

IN THERMAL EQUILIBRIUM:IN THERMAL EQUILIBRIUM:

TTCOORDCOORD == TTKINKIN == TT

We may consider furtherWe may consider furtheronly potential energy:only potential energy:

EE EECOORDCOORD

MM MMCOORDCOORD

S(E)S(E) SSCOORDCOORD(E(ECOORD COORD ))

F(E)F(E) FFCOORDCOORD , etc. , etc.

TRANSITIONS:TRANSITIONS:THERMODYNAMICSTHERMODYNAMICS

gradual transitiongradual transition

““all-or-none” (or first order) all-or-none” (or first order) phase transitionphase transition

coexistencecoexistence& jump& jump

coexistencecoexistence

((T/TT/T**)()(E/kTE/kT**)) ~~ 11

““all-or-none” (or first order) phase transitionall-or-none” (or first order) phase transition

F(T1)

Second order phase transitionSecond order phase transition

changechange

Not observed in proteins up to now: Not observed in proteins up to now: they are too smallthey are too small

TRANSITIONS:TRANSITIONS:KINETICSKINETICS

nn## == nn exp(-exp(-FF##/k/kBBT)T)nn# #

nn

TRANSITION TIME:TRANSITION TIME:

tt0011 = = tt00#1#1 = =

= = ## (n/n (n/n##)) == ## exp(+exp(+FF##/k/kBBT)T)

PARALLEL REACTIONS:PARALLEL REACTIONS:

TRANSITION TRANSITION RATERATE = SUM OF = SUM OF RATESRATES (or: (or: the highest rate)the highest rate)

RATERATE = 1/ = 1/ TIMETIME

CONSECUTIVE REACTIONS:CONSECUTIVE REACTIONS:

TRANSITION TRANSITION TIMETIME SUM OF SUM OF TIMESTIMES

tt00… … finish finish = = tt00#1#1 finish finish + + tt00#2#2 finish finish + … + …

## ##

startstart finishfinish

__

__ __

TRANSITION TIME IS ESSENTIALLY TRANSITION TIME IS ESSENTIALLY

EQUAL FOR “TRAPS” EQUAL FOR “TRAPS” ATAT AND AND OUT OFOUT OF PATHWAYS OF CONSECUTIVE REACTIONS:PATHWAYS OF CONSECUTIVE REACTIONS:

TRANSITION TRANSITION TIMETIME SUM OF SUM OF TIMESTIMES

(or: (or: the longest time)the longest time)

## mainmain

finishfinish finishfinishstartstartstartstart

““trap”: ontrap”: on ““trap”: outtrap”: out

## mainmain

DIFFUSION:DIFFUSION:KINETICSKINETICS

Mean kinetic energy of a particle:Mean kinetic energy of a particle: mvmv22/2/2 ~ k ~ kBBTT

<<>> = = j j PPjj((jj)) jj v v22 = (v = (vXX22)+(v)+(vYY

22)+(v)+(vZZ22))

Maxwell:Maxwell:

Friction stops a molecule within picoseconds:Friction stops a molecule within picoseconds: m(dv/dt) = -(3m(dv/dt) = -(3DD)v )v [Stokes law][Stokes law]D – diameter; D – diameter; m ~ Dm ~ D33 – mass; – mass; – – viscosityviscosity

ttkinetkinet 10 10-13 -13 sec sec (D/nm) (D/nm)22 in waterin water

During tDuring tkinetkinet the molecule moves somewhere by the molecule moves somewhere by LLkinetkinet ~ v ~ v••ttkinetkinet

Then it restores its kinetic energy mvThen it restores its kinetic energy mv22/2 ~ k/2 ~ kBBT from thermal T from thermal

kicks of other molecules, and moves in another random sidekicks of other molecules, and moves in another random side

CHARACTERISTIC DIFFUSION TIME: nanosecondsCHARACTERISTIC DIFFUSION TIME: nanoseconds

Friction stops a molecule within picoseconds:Friction stops a molecule within picoseconds: ttkinetkinet 10 10-13 -13 sec sec (D/nm) (D/nm)22 in waterin water

During During ttkinetkinet the molecule moves somewhere by the molecule moves somewhere by LLkinet kinet ~~ vv••ttkinetkinet

Then it restores its kinetic energyThen it restores its kinetic energymvmv22/2 ~ k/2 ~ kBBT from thermal kicks T from thermal kicks

of other molecules, and moves in of other molecules, and moves in another random sideanother random side

CHARACTERISTIC DIFFUSION CHARACTERISTIC DIFFUSION TIME: nanosecondsTIME: nanoseconds

The random walk allows the molecule The random walk allows the molecule to diffuse at distance D (~ its diameter) to diffuse at distance D (~ its diameter) within ~within ~(D/L(D/L kinet kinet))22 steps, i.e., withinsteps, i.e., within

ttdifftdifft t tkinetkinet•(D/L•(D/Lkinetkinet))22 4 4•1•100-10 -10 sec sec (D/nm) (D/nm)33 in waterin water

For “small part”: For “small part”: PPjj((jj) = exp(-) = exp(-jj/k/kBBT)/Z(T); T)/Z(T);

Z(T) = Z(T) = j j exp(-exp(-jj/k/kBBT)T)

j j PPjj((jj) = 1) = 1

E(T) = E(T) = <<>> = = j j jj PPjj((jj) )

if allif all j j == : #: #STATESSTATES = 1/P, i.e.: S(T) = k = 1/P, i.e.: S(T) = kBBln(1/P)ln(1/P)

S(T) = kS(T) = kBB<<ln(#ln(#STATESSTATES))>> = k= kBBj j ln[1/Pln[1/Pjj((jj)])]PPjj((jj) )

F(T) = E(T) - TS(T) = -kF(T) = E(T) - TS(T) = -kBBT T ln[ln[ Z(T)] Z(T)]

STATISTICAL MECHANICSSTATISTICAL MECHANICS

Thermostat: Thermostat: TTth th = dE= dEthth/dS/dSthth

““Small part”: PSmall part”: Pjj((jj,T,Tthth) ~ exp(-) ~ exp(-jj/k/kBBTTthth););

E(TE(Tthth) = ) = j j jj PPjj((jj,T,Tthth) )

S(TS(Tthth) = k) = kBBj j ln[1/Pln[1/Pjj((jj,T,Tthth)])]PPjj((jj,T,Tthth) )

TTsmall_partsmall_part = dE(T = dE(Tthth)/dS(T)/dS(Tthth) = T) = Tthth

STATISTICAL STATISTICAL MECHANICSMECHANICS

Along tangent: S-S(EAlong tangent: S-S(E11)) = (E-E= (E-E11)/)/ TT11

i.e.,i.e.,

F = E - TF = E - T11SS = = constconst (= F (= F11 = E = E11 - T - T11SS11) )

Separation of potential energySeparation of potential energyin classic (non-quantum) mechanics:in classic (non-quantum) mechanics:

P(P() ~ exp(-) ~ exp(-/k/kBBT) T) Classic:Classic: == COORDCOORD + + KINKIN

KINKIN = mv= mv22/2 : /2 : does not depend on coordinatesdoes not depend on coordinates

Potential energyPotential energy COORDCOORD: : depends only on coordinatesdepends only on coordinates

P(P() ~ exp(-) ~ exp(-COORDCOORD/k/kBBT) T) • • exp(-exp(-KINKIN/k/kBBT)T)

ZZKINKIN(T) =(T) = K K expexp(-(-KK/k/kBBT): T): don’t depend on coord.don’t depend on coord.

ZZCOORDCOORD(T) =(T) = CCexpexp(-(-CC/k/kBBT): T): depends on coord.depends on coord.

Z(T) = ZZ(T) = ZCOORDCOORD(T)(T)••ZZKINKIN(T) (T) F(T) = F F(T) = FCOORDCOORD(T)(T)+F+FKINKIN(T) (T) ================================================================================================================================================================================================================================================

Elementary volume: Elementary volume: (mv)(mv)x = x = ħ ħ ( (x)x)33 =( =(ħ/|mv|)ħ/|mv|)33

P(P(KINKIN++COORDCOORD) ~ exp(-) ~ exp(-COORDCOORD/k/kBBT)T)••exp(-exp(-KINKIN/k/kBBT)T)

P(P(COORDCOORD) = exp(-) = exp(-COORDCOORD/k/kBBT) / ZT) / ZCOORDCOORD(T)(T)

ZZCOORDCOORD(T) =(T) = CCexpexp(-(-CC/k/kBBT): T): depends depends ONLY ONLY

on coordinateson coordinates

P(P(KINKIN) = exp(-) = exp(-KINKIN/k/kBBT) / ZT) / ZKINKIN(T)(T)

ZZKINKIN(T) =(T) = K K expexp(-(-KK/k/kBBT): T): don’t depend on coord.don’t depend on coord.

T<0: unstable (explodes)T<0: unstable (explodes)

<<KINKIN> > at T<0 at T<0

due todue to

P(P(KINKIN) ~ exp(-) ~ exp(-KINKIN/k/kBBT)T)