Theoretical and computational physical chemistry group
Theoretical characterization of Theoretical characterization of -helix -helix
and and -hairpin folding kinetics-hairpin folding kinetics
Isabella Daidone
Cambridge, 26/07/2005
Theoretical and computational physical chemistry group
Free energy
ProteinProtein foldingfolding
Theoretical and computational physical chemistry group
Folding kineticsFolding kinetics
[ms - minutes] [s timescale]
Proteins
Peptides
[10 ns – hundreds ns] [hundreds ns - s]
C-terminal -hairpin of GB1 protein 6 speptide I 760 ns
alanine-rich peptides
Theoretical and computational physical chemistry group
Folding kineticsFolding kinetics
folding kinetics is still computationally prohibitive !!!
Peptides
[10 ns – hundreds ns][hundreds ns - s]
Molecular Dynamics (MD) simulations
[hundreds of ns]
•Simplified protein models
•Massive parallel computing
•Transition path sampling
•Diffusive rate theory
Theoretical and computational physical chemistry group
( ) ( , ) ( )k kP t q t q dqr c= ò
( , ) ( , ) ( ) ( , )q t q t A q q tD
t q q q RT
(q,t), peptide probability density
Pk(t), time probability of secondary structure states, k
equilibrium fraction of secondary structure states, k
( )
7( )
0
( )k
k
A q
kA q
k
eq
e
b
bc
- D
- D
=
=
å
( )( ) ln
( )eq
eq ref
qA q RT
qr
rD = - external potential given
by the free energy
A (
q)
folding reaction coordinate, q
Theoretical and computational physical chemistry group
MD simulation
• Starting structure• Equations of motion (force field)
Kinetic simulation
• Starting condition of the system• Diffusion equations (free energy gradient)
A (
kJ/
mol)
q
Theoretical and computational physical chemistry group
Folding kinetic simulationFolding kinetic simulation
• Proper reaction coordinates, q
• Free energy profile, A(q) , and k(q)
• Diffusion coefficient, D
Theoretical and computational physical chemistry group
Folding kinetic simulationFolding kinetic simulation
• Proper reaction coordinates, q
• Free energy profile, A(q) , and k(q)
• Diffusion coefficient, D
Theoretical and computational physical chemistry group
““Reaction coordinates”: essential degrees of Reaction coordinates”: essential degrees of freedomfreedom
A. Amadei et al., PROTEINS, 17:412-425, 1993. Essential dynamics of proteins
Theoretical and computational physical chemistry group
““Reaction coordinates”: essential degrees of Reaction coordinates”: essential degrees of freedomfreedom
A. Amadei et al., PROTEINS, 17:412-425, 1993. Essential dynamics of proteins
ij i i j j C x x x x Positional fluctuationsPositional fluctuationscovariance matrixcovariance matrix
Eigenvectors of fluctuationsEigenvectors of fluctuationsand corresponding eigenvaluesand corresponding eigenvaluesCTTΛ T
q first essential eigenvector
Theoretical and computational physical chemistry group
Folding kinetic simulationFolding kinetic simulation
• Proper reaction coordinates, q
• Free energy profile, A(q) , and k(q)
• Diffusion coefficient, D
Theoretical and computational physical chemistry group
Thermodynamic propertiesThermodynamic properties
( ) ( )( ) ln ln
( ) ( )eq
eq ref ref
q N qA q RT RT
q N qr
rD = - -;
( )
7( )
0
( )( )
( )
k
k
A qk
kA q
k
N qeq
N qe
b
bc
- D
- D
=
=
å;
A (
q)
q
Theoretical and computational physical chemistry group
Folding kinetic simulationFolding kinetic simulation
• Proper reaction coordinates, q
• Free energy profile, A(q) , and k(q)
• Diffusion coefficient, D
Theoretical and computational physical chemistry group
Free diffusion over the essential Free diffusion over the essential eigenvectoreigenvector
time (ns)
q (
nm
)
-3-3
-2-2
-1-1
00
11
Theoretical and computational physical chemistry group
D and D0 are the long and short-time diffusion constants, respectively
1 , 2, 3 are the relaxation times of the two switching modes
mean square displacement <q2(t)> (nm2/ps)
<q2(t)> = 2 t + 2 (D0-A1) 1 (1-exp(-t/1)) + 2 (D0-A2) 2 (1-exp(-t/2))
+ 2 (D0-A3) 3 (1-exp(-t/3))
D
Free diffusion over the essential Free diffusion over the essential eigenvectoreigenvector
slope=2D0
slope=2D00
Theoretical and computational physical chemistry group
Model systemsModel systems
Prion Protein H1 peptide (14 residues)
Antimicrobial temporin L (13 residues)
MKHMAGAAAAGAVV FVQWFSKFLGRIL
Amyloidogenic(-sheet rich firbrils)
-helical(circular dichroism)
Theoretical and computational physical chemistry group
MD simulationsMD simulations* of the H1 of the H1 peptidepeptide
•PME•N,V,T (isokinetic)
•periodic truncated octahedron•1 nm explicit solvent on all sides
*GROMACS software package
Total simulation time of 1.1 s
850300 water (SPC)-hairpin
240300water (SPC)-helix
Length (ns)Temp (K)SolventStarting structure
Theoretical and computational physical chemistry group
MD simulationsMD simulations* of temporin Lof temporin L
300300water (SPC)-helix
Length (ns)Temp (K)SolventStarting structure
•PME•N,V,T (isokinetic)
•periodic truncated octahedron•1 nm explicit solvent on all sides
*GROMACS software package
Theoretical and computational physical chemistry group
1
H1 peptideH1 peptide
+ + + + + + = 30% = 70%
Theoretical and computational physical chemistry group
5
temporin Ltemporin L
280240
Time (ns)
= 16%= 14%= 70%
Theoretical and computational physical chemistry group
A (
kJ/
mol)
q (nm) q (nm)
0 1 2 3 4 5 76 L=0 L=1 L=2
k q
L=1
L=0
L=2
Theoretical and computational physical chemistry group
Diffusion coefficientsDiffusion coefficients
2.4 10-5
(0.5 10-5)
-hairpin
3.2 10-5
(0.5 10-5)
-helix
Dnm2ps-1
system
<1
<1
1ps
0.026(0.001)
0.026(0.001)
D0nm2ps-1
7.8(1)
4.3(1)
2ps
113.4(2)
30.6(2)
3ps
Theoretical and computational physical chemistry group
Eigenvector I (nm)
(q,t)0
50
150
100
Tim
e (n
s)
0
1
-1-2
200
( ) ( , ) ( )k kP t q t q dqr c= òA
(kJ/
mol)
q (nm)
( , ) ( , ) ( ) ( , )q t q t A q q tD
t q q q RT
Theoretical and computational physical chemistry group
-hairpin mean folding time-hairpin mean folding time
fold 30 ns-helix mean folding time-helix mean folding time
fold 8 ns
Pk(q,t)Pk(q,t)
Time (ns)
+ + + + + +
Theoretical and computational physical chemistry group
s
V. Munoz et al., Nature, Folding dynamics and mechanism of -hairpin formation. 390:196-199, 1997.
TerminalTerminal-hairpin of Protein G-hairpin of Protein G
exp s
A# kJ/molA
(kJ/
mol)
q (nm)
Theoretical and computational physical chemistry group
q (nm)
A (
kJ/
mol)
-hairpin folding kinetics-hairpin folding kinetics
bi
one
tt
= 1.22
Daidone et al., 2005. Theoretical characterization of -helix and -hairpin folding kinetics, Manuscript in
preparation.
q1 (nm)q
2 (
nm
)
Theoretical and computational physical chemistry group
AcknowledgmentsAcknowledgments
Dr. Marco D’Abramo
Prof. Jeremy Smith (University of Heidelberg)
Prof. Alfredo Di Nola(University “La Sapienza” of Rome)
Dr. Andrea Amadei (University of Rome “Tor Vergata” )
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