HHMI meeting, FOLDING

1
PROTEIN PHYSICS LECTURE 9
2
Secondary structure transitions We may consider
further only potential energy E ? ECOORD M ?
MCOORD S(E) ? SCOORD(ECOORD ) F(E) ? FCOORD ,
etc.
3
?-helix
homo-polypeptide ?F? F? - Fcoil
(n-2)fH - nTS? -2fH n?(fH - TS?)


fINIT fEL s exp(-fEL/kBT)
elongation (S ? 1)
?2 Ala -- ?0.2 Gly ?
exp(-fINIT/kBT) initiation (?ltlt1)

0.001 Pn_? /Pn_COIL exp(-?F?/kBT) ? sn
4
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
?F?_n Const n?f ?FICE_n
C?n2/3 n?f interface
interface
Mid-transition
f 0 ?S?_n ln(N)
?SICE_n ln(N) N
very large Const ltlt ln(N)
N2/3 gtgt ln(N) phases mix
phases do not mix

5
Average lengths n0 of helix and coil regions at
mid-transition (when fEL0)
N
of ends ? N/n 1 ? N/n of states
N(N-1)(N-?1) / ?(?-1)1 S(?)/kB
lnN(N-?1) / ?1 ?S/kB S(?1) -
S(?)/kB ln(N-?)/(?) lnN/?-1 E(?)
(?/2)fINIT (when fEL0) ?E E(?1) - E(?)
fINIT/2 EQUILIBRIUM ?G ?E-T?S0 fINIT/2 -T
kBlnN/?-1 0 ? n0-1 exp(fINIT/2kBT)
?-1/2
n0 ? ?-1/2 (when ? ltlt1)
n
n
6
Width of helix-coil transition When fEL
changes IF n0 ?fEL lt - kBT, i.e., fEL/kBT lt
- 1/n0 stable helix IF n0 ?fEL gt kBT
i.e., fEL/kBT gt 1/n0 unstable helix,

stable coil
n0
Transition width ? fEL/kBT 2?1/2
fEL0 if ? 50 for very long chain n0 ?
? 0 when chain is shorter than n0
n0 ? ?-1/2 ? 30
7
TIME of coil-helix transition Barrier for
initiation ?F fINIT Time to initiate helix
in given place t1 ? ? exp(?F/kBT) ? ? ?
-1 ? 110 ns Time to initiate helix
in any of n0 places tINIT_H n0-1 ? ? ?
exp(?F/kBT) ??? -1/2 100 ns To extend helix
to n0 residues tEL_H n0 ? ? ? ? ? -1/2
100 ns tHELIX 200
ns
n0
/
8
TIME of coil stable ?-hairpin
transition Barrier for initiation ?F fTURN ?
fINIT_? Time to initiate ?-hairpin with turn
in the middle of the chain t1 ? ?
exp(?F/kBT) ? ? ? -1 3000 ns Time to
extend ?-hairpin to n residues tEL_?-HAIRPIN
n ? ? 100 ns t?-HAIRPIN 3000 ns
fTURN
n
/
9
TIME of coil ?-sheet transition (when hairpin
is unstable)
fTURN
fEDGEf?
f? lt 0 fEDGE f? gt 0
f?
F
f?N fTURN lt 0 ? Nmin fTURN/(-f?)
F fTURN 2Nmin(fEDGEf?) fTURN 2 fTURN
fEDGE /(-f?)
10
TIME of coil ?-sheet transition
F 2 fTURN fEDGE /(-f?) ? ? when (-f?) ? 0
Time to initiate ?-sheet folding t1 ? ?
exp(?F/kBT) ? ? when (-f?) ? 0 !!
Fopt(M) 2 fTURN fEDGE /(-f?) - fTURN
fTURN
fEDGEf?
F
f?
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coil
r ________ L
1cos?
1 cos?
lth2gt (Mr)r LMr h M1/2

V M3/2