PROTEIN PHYSICSPROTEIN PHYSICS
LECTURES 7-8LECTURES 7-8
Basics of thermodynamics & kineticsBasics of thermodynamics & kinetics
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 •• MMtt(E-(E-) )
SStt(E-(E-) ) k k •• ln[Mln[Mtt(E-(E-)])]
SStt(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)) // M(E)M(E) = =
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 → → inf.inf.) Unstable (falls)) Unstable (falls)
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 transition
“all-or-none” (or 1st order) phase transition
coexistence& jump-liketransition
coexistence
((E/kTE/kT**)()(T/T*)T/T*) ~~ 11Transition: |F1|= |-ST| ~ kT*
E-T*S=0
Second order phase transitionSecond order phase transition
changechange
Recently observed in proteins;Recently observed in proteins;they are rarethey are rare
LANDAULANDAU:: Helix-coil transition: Helix-coil transition: Melting:Melting:NOT 1-s order phase transition NOT 1-s order phase transition 1-s order phase transition1-s order phase transition
Helix & coil: 1D objects Helix & coil: 1D objects Ice & water: 3D objectsIce & water: 3D objects
NNNN
nnnn
rule of contraries
FFhelix_nhelix_n = = ConstConst + n + nf f FFICE_nICE_n = = CCnn2/32/3 + n + nf f
1D interface 3D interface 1D interface 3D interface
Mid-transition: Mid-transition: f f = 0= 0
SShelix_nhelix_n ~ ln(N) ~ ln(N) positions positions SSICE_nICE_n ~ ln(N) ~ ln(N) N : very large; n ~ N : very large; n ~ N, N, <<1 (<<1 (~0.001)~0.001) Const << ln(N)Const << ln(N) 2/32/3NN2/32/3 >> ln(N) >> ln(N)
phases mixphases mix phases do not mix phases do not mix
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)
Not Not ““slowly goes”,slowly goes”,butbutclimbs, falls climbs, falls and climbs again…and climbs again…
fallsfalls
#
TRANSITION TRANSITION RATERATE = = SUM OF SUM OF RATESRATES (or: (or: the highest rate)the highest rate)
1/TIME = (1/1/TIME = (1/##) ) exp(- exp(-FF11##/k/kBBT) + (1/T) + (1/##) ) exp(- exp(-FF22
##/k/kBBT)T)
PARALLEL REACTIONS:PARALLEL REACTIONS:
RATERATE = 1/ TIME = 1/ TIME
tt00… … finish finish = = tt00#1#1 finish finish + + tt00#2#2 finish finish + … + …
## ##
startstart finishfinish
__ CONSECUTIVE REACTIONS:CONSECUTIVE REACTIONS: TRANSITION TRANSITION TIMETIME SUM OF SUM OF TIMESTIMES
(or: (or: the highest time)the highest time)
TIME = TIME = ## exp(+exp(+FF11##/k/kBBT) + T) + ## exp(+exp(+FF22
##/k/kBBT) + …T) + …
__ __
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:
in 3Din 3D
Friction stops a molecule within Friction stops a molecule within picosecondspicoseconds:: m(dv/dt) = -(3m(dv/dt) = -(3DD)v )v [Stokes law][Stokes law]D – diameter; D – diameter; m ~ Dm ~ D33 1g/cm 1g/cm3 3 – 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: CHARACTERISTIC DIFFUSION TIME: nanosecondsnanoseconds
Friction stops a molecule within Friction stops a molecule within picosecondspicoseconds:: ttkinetkinet 10 10-13 -13 sec sec (D/nm) (D/nm)22 in waterin water
DIFFUSION:DIFFUSION: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: TIME: nanosecondsnanoseconds
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)
““all-or-none” (or first order) phase transitionall-or-none” (or first order) phase transition
F(T1)
________________________________