exploring landscapes .
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700 K replica. energy. 200 K replica. “important coordinates”. Exploring landscapes. Exploring landscapes for protein folding and binding using replica exchange simulations. Effective potential. Important coordinates. The AGBNP all atom effective solvation potential & REMD - PowerPoint PPT PresentationTRANSCRIPT
Exploring landscapes . . .
“important coordinates”
ener
gy
700 K replica
200 K replica
Important coordinates
Eff
ecti
ve p
oten
tial
Exploring landscapes for protein folding and binding using replica exchange simulations
• The AGBNP all atom effective solvation potential & REMD
• Peptide free energy surfaces & folding pathways from all atom simulations and network models
• Temp. dependence of folding: physical kinetics and replica exchange kinetics using network models
• Replica exchange on a 2-d continuous potential with an entropic barrier to folding
AGBNP effective solvation potential(Analytical Generalized Born + Non Polar)
Gsolv Gelec Gnp
Gcav GvdW
• OPLS-AA AGBNP effective potential, an all atom model• Novel pairwise descreening Generalized Born model.• Separate terms for cavity free energy and solute-solvent van der
Waals interaction energy.• Fully analytical.• Applicable to small molecules and macromolecules.
Generalized BornSurface area model Born radius-based estimator
E. Gallicchio, and R.M. Levy, JCC, 25, 479 (2004)
AGBNP: Pairwise Descreening Scheme
i
Born radii: rescaled pairwise descreening approximation:
1
Bi
1
Ri 1
4 s jQijj
Rescale according to self-volume of j:
s j Vj (self)
Vj
Vj (self)Vj 1
2 Vjkk 1
3 Vjklkl
Self-volume of j (Poincarè formula, ca. 1880):
E. Gallicchio, R. Levy, J. Comp. Chem. (2004)Hawkins, Cramer, and Truhlar, JPC 1996Schaefer and Karplus, JPC 1996Qiu, Shenkin, Hollinger, and Still, JPC 1997
j
Non-Polar Hydration Free Energy
Gnp i Ai iW(Bi ) i
Non-polar hydration free energy estimator:
Gnp Gcav GvdW
Wi w-4i i
6
| r ri |6slv. 16wi i
6
3Ci3
Ci 3
41
| r - ri |6slv.
1/ 3
Bi
: Surface area of atom i
: Estimator based on Born radius
: Surface tension and van der Waals adjustable parameters
Ai
W(Bi )
i ,i
R.M. Levy, L. Y. Zhang, E. Gallicchio, and A.K. Felts, JACS, 125, 9523 (2003) (proteins in water)
E. Gallicchio, M. Kubo, and R.M. Levy, JPC, 104, 6271 (2000) (hydrocarbons in water)
Enthalpy-Entropy and Cavity Decomposition of Alkane Hydration Free Energies: Numerical Results and
Implications for Theories of Hydrophobic Solvation
E. Gallicchio, M. Kubo, R. M. Levy, J. Phys. Chem., 104, 6271 (2000)
The replica exchange method for structural biology problems
• Has been successfully applied to protein and peptide folding, ligand binding, and NMR structure determination
• Questions have been raised about the efficiency of the algorithm relative to MD
e.g. Nymeyer, Gnanakaran & García (2004) Meth. Enz. 383: 119 Ravindranathan, Levy, et al. (2006) JACS 128: 5786 Chen, Brooks, et al. (2005) J. Biomol. NMR 31: 59
Beck, White & Daggett (2007) J. Struct. Biol. 157: 514Zuckerman & Lyman (2006) JCTC 2: 1200 (with erratum)
Replica exchange molecular dynamics
rough energy landscapes and distributed computing
200 K
MD MD MD MD MD700 K
“important coordinates”
ener
gy
450 K
320 K
Y. Sugita, Y. OkamotoChem. Phys. Let., 314,261 (1999)
Replica exchange molecular dynamics
rough energy landscapes and distributed computing
200 K
MD700 K
450 K
320 K
Y. Sugita, Y. Okamoto (1999) Chem. Phys. Let., 314:261
“important coordinates”
ener
gy700 K replica
200 K replicawalker 4
walker 1
repl
ica
MD MD MD MD MD MD
walker 2
walker 3
Protein folding: REM and kinetic network models
• free energy surfaces of the GB1 peptide from
REM and comparison with experiment
• kinetic network model of REMD
(simulations of simulations)
F2 U2
F1 U1
Andrec M, Felts AK, Gallicchio E, Levy RM.. PNAS (2005) 102:6801.
• kinetic network model of folding pathways
for GB1
Zheng W, Andrec M, Gallicchio E, Levy RM. PNAS (2007) 104:15340.
The -Hairpin of B1 Domain of Protein G
Folding nucleus of the B1 domain Blanco, Serrano. Eur. J. Biochem. 1995, 230, 634.
Kobayashi, Honda, Yoshii, Munekata. Biochemistry 2000, 39, 6564.
Features of a small protein: stabilized by 1) formation of secondary structure 2) association of hydrophobic residues Munoz, Thompson, Hofrichter, Eaton. Nature 1997, 390, 196.
Computational studies using Explicit and Implicit solvent models Pande, PNAS 1999 Dinner,Lazaridis,Karplus,PNAS,1999 Ma & Nussinov, JMB, 2000 Pande, et al., JMB, 2001 Garcia & Sanbonmatsu, Proteins, 2001 Zhou & Berne, PNAS, 2002
The -Hairpin of B1 Domain of Protein G
The potential of mean force of the capped peptide.
Simple (surf area) nonpolar model OPLS/AGBNP
A Felts, Y. Harano, E. Gallicchio, and R. Levy, Proteins, 56, 310 (2004)
-hairpin > 90%-helix < 10%G ~ 2 kcal/mol
Kinetic network models for folding
Network nodes are snapshots from multiple temperatures of a replica exchange simulation.
• Waiting time in a state is an exponential random variable with mean = 1/(j kij)• Next state is chosen with probability proportional to k ij
Simulations are performed using the Gillespie algorithm for simulating Markov processes on discrete states:
Transition rates (edges) are motivated by Kramers theory: transitions are allowed if there is sufficient structural similarity, and forbidden otherwise.
Dynamical/kinetic considerations:
Equilibrium considerations:Sufficiently long trajectories must reproduce WHAM results.
800,000 nodes7.4 billion edges
Tcold Thot
Andrec, Felts, Gallicchio & Levy (2005) PNAS, 102, 6801
Connection between kinetic model and equilibrium populations
Equilibrium populations for temperature T0 are preserved if for each pair of nodes (i, j) the ratio of transition rates follows WHAM weighting:
node i from temperature TAhaving energy Ei
node j from temperature TB having energy Ej
where fA(0) and fB(0) are free energy weights for the TA and TB simulations at reference temperature T0
These weights are order-parameter independent and will give correct PMFs for any projection.
T-WHAM PMF at low temperature contains information from high temperature simulations
The majority of beta-hairpin folding trajectories pass through alpha helical intermediate states
91% of 4000 temperature-quenched stochastic trajectories begun from high-energy coil states pass through states with -helical content
Fraction of hairpin conformation averaged over 4000 stochastic trajectories run at 300 K and begun from an initial state ensemble equilibrated at 700 K.
= 2500 units ≈ 50 µs = 9 units ≈ 180 ns
Andrec, Felts, Gallicchio & Levy (2005) PNAS, 102, 6801
• Myoglobin and coiled-coil proteins can form amyloid fibrils
Evidence for -helical intermediates in -sheet folding and misfolding
• Non-native helices have been observed in -lactoglobulin folding• Rapid formation of structure• Can exist as a stable thermodynamic species and as intermediates• May be important in protecting exposed ends of -sheet from intermolecular
interactions
Kirkitadze, Condron & Teplow (2001) JMB 312:1103Fezoui & Teplow (2002) JBC 277: 36948
Forge, Hoshino, Kuwata, Arai, Kuwajima, Batt & Goto (2000) JMB 296:1039Kuwata, Shastry, Cheng, Hoshino, Batt, Goto & Roder (2001) Nat. Struct. Biol. 8:151
• Amyloid -sheets can form from -helical precursors
Fändrich, Forge, Buder, Kittler, Dobson & Diekmann (2003) PNAS 100:15463Kammerer, Dobson, Steinmetz et al. (2004) PNAS 101: 4435
• Entropy-stabilized helical intermediates may be generic in -sheet protein folding landscapes
• Computational and theoretical evidence
García & Sanbonmatsu (2001) Proteins 42:345Zagrovic, Sorin & Pande (2001) JMB 313:151Wei, Mousseau & Derreumaux (2004) Proteins 56:464
Chikenji & Kikuchi (2000) PNAS 97:14273
• Helical structures have been observed in G-peptide simulations
• Fibril formation in amyloid -protein may occur via a helical intermediate
Important coordinates
Eff
ecti
ve p
oten
tial
Exploring landscapes for protein folding and binding using replica exchange simulations
• The AGBNP all atom effective solvation potential & REMD
• Peptide free energy surfaces & folding pathways from all atom simulations and network models
• Temp. dependence of folding: physical kinetics and replica exchange kinetics with a network model
• Replica exchange on a 2-d continuous potential with an entropic barrier to folding
Network models of Replica Exchange
F Uku
kf
F1U2
U1F2
U1U2
F1F2
F2U1
U2F1
U2U1
F2F1
One walker Two walkers
F2 U2
F1 U1
2 walkers: 8 states
N walkers
F2 U2
F1 U1
FN UN
5 walkers: 3840 statesN walkers: 2N N! states
Gillespie “simulation of protein folding simulations”
kRE kRE
ku2
kf2
ku1
kf1
kRE
ku1
kf1
ku2
kf2
kuN
kfN
ku and kf: physical kineticskRE: replica exchange “kinetics”
Convergence at low temperature depends on the number of F1 to U1 to F1 “transition events”
Speed limit for replica exchange efficiency
†
†
Temperature of high-temperature replica T2 (K)
Number of transition events at low temperature T1 in 1 ms
Arrhenius case (∆Cp
† = 0)
Non-Arrhenius case (∆Cp
† < 0)
300 111 111
440 1049 532
700 2801 134
The number of transition events at low temperature is approximately equal to the average of the harmonic means of the rate constants at all temperatures:
Results for 2 walkers:
Non-Arrhenius case (∆Cp
† < 0)
Replica exchange convergence is dependent on the physical kinetics of
the system
• The number of transition events depends on the average of the harmonic mean rates, and sets a “speed limit” for efficiency
• Maximizing the rate of temperature diffusion is appropriate if the underlying kinetics is Arrhenius
• For non-Arrhenius kinetics, an optimal temperature exists which maximizes the number of transition events and convergence
• “Training” simulations (like those used for the multicanonical method) may be useful to locate optimal maximal temperatures
Zheng W, Andrec M, Gallicchio E, Levy RM. PNAS (2007) 104:15340.
2-d continuous potential
Potential energy along x
F U
Replica exchange on a 2-d continuous potential with an entropic barrier to folding
Simple Continuous and Discrete Models for Simulating Replica Exchange Simulations of Protein Folding W. Zheng, M. Andrec, E. Gallicchio, R. M. Levy, J. Phys. Chem., in press
s
Rate constants extracted from (Uncoupled) simulations on the 2-d continuous potential
ku
kf
fex= 10-2fex= 10-4
0.430.420.42ku(T2)
0.310.290.30kf(T2)
0.00370.00380.0036ku(T1)
6.36.46.1kf(T1)
Reverse-engineering ratesUncoupled
Reverse-Engineering rates from the trajectory on the continuous potential using lifetime & branching
ratiosT1=296K
T2=474K
3
1
1
jj
ii
branch
branchk
RE on the continuous potential vs RE on the kinetic network
Infinitely fast exchange limit*
The faster the replica exchange rate, the bigger the discrepancy.
Kinetic network
Continuous potl
fex = 5·10-3
fex = 1·10-3
fex = 5·10-2
*Calculated using harmonic mean of rate constants
fex =5·10-3
0.8860.150P(U2F1F2F1)
0.8950.477P(F2F1 U2F1F2F1)
0.0940.521P(F2F1 U2F1U1F2)
0.1030.849P(U2F1U1F2)
Calculated from the continuous traj. at
different exchange ratesProbability
Non-Markovian effects -- History dependence
fex =10-4
• Non-Markovian effects are observed in Replica Exchange simulations on the continuous potential
• When the frequency of replica exchange exceeds the
time scale for relaxation in the F and U macrostates, the convergence rate slows
• The efficiency of RE in more complex systems is fundamentally limited by the time scale of conformational diffusion within the free energy basins.
Summary
Important coordinates
Eff
ecti
ve p
oten
tial
Exploring landscapes for protein folding and binding using replica exchange simulations
• The AGBNP all atom effective solvation potential & REMD
Emilio Gallicchio• Peptide free energy surfaces & folding pathways from all atom simulations and network models
Tony Felts, Zenmei Ohkubo, and Michael Andrec• Temp. dependence of folding: physical kinetics and replica exchange kinetics
Weihua Zheng, Michael Andrec, Emilio Gallicchio• Replica exchange on a 2-d continuous potential with an entropic barrier to
folding Weihua Zheng, Michael Andrec, Emilio Gallicchio
Important coordinates
Eff
ecti
ve p
oten
tial
Protein Folding with All Atom Potentials Insights using Replica Exchange and Network
Models
• The AGBNP all atom effective solvation potential Emilio Gallicchio, Tony Felts
• Peptide free energy surfaces and folding pathways Tony Felts, Zenmei Ohkubo, and Michael Andrec
• Network models and kinetics in the replica exchange ensemble Michael Andrec, Emilio Gallicchio
Potential of mean force (PMF) along x
F U
Replica exchange convergence is dependent on the physical kinetics of
the system
• The number of transition events depends on the average of the harmonic mean rates, and sets a “speed limit” for efficiency
• Maximizing the rate of temperature diffusion is appropriate if the underlying kinetics is Arrhenius
• For non-Arrhenius kinetics, an optimal temperature exists which maximizes the number of transition events and convergence
• “Training” simulations (like those used for the multicanonical method) may be useful to locate optimal maximal temperatures
Replica Exchange and Ligand Binding
• Binding free energy landscape contains multiple minima
• Effect of binding & temperature is to shift distribution of conformations
• Replica Exchange addresses the sampling problem while providing estimates of populations
Folding Landscape Binding Landscape
Reduced Coordinate
The P450 puzzle
• Several X-ray crystal structures of P450s show substrate distant from active site
• Hypothesized a conformational equilibrium between productive and unproductive conformational states
• Cytochrome P450s metabolize many aliphatic molecules and 90% of pharmaceutical ligands
NPG
Heme
Phe87
P450BM-3/NPG
Experimental and Modeling Clues
Induced Fit docking finds a Fe-bound conformation of higher energy than the X-ray conformation
X-ray (Distal)
Induced Fit Model* (Proximal)
ω1-Fe
• UV-VIS and SSNMR experiments indicate temperature-dependent equilibrium between Fe-bound and un-bound species.
*Jovanovic, T.; Farid, R.; Friesner, R. A.; McDermott, A. E. J. Am. Chem. Soc. 2005, 127, 13548.
Questions:• Do the Xray and Induced Fit structures correspond to the low and
high temperature conformations? • Are there other states?• What’s the mechanism of interconversion between states?
REMD of P450 NPG Complex
• Model ligand and 120 active site residues
• 24 replicas between 260 and 463 K
• 72 ns total aggregate simulation time
• Provides populations of conformational states (canonical sampling) as a function of temperature
• Can be used to construct free energy landscape
Ravindranathan, K.P., E. Gallicchio, R.A. Friesner, A.E. McDermott, and R.M. Levy. J. Am. Chem. Soc., 128, 5786-5791 (2006).
• New proximal ligand-free state, most populated at physiological temperature. Entropically stabilized
• Conversion from distal state goes through proximal ligand-locked conformation
• Barrier from proximal to distal is about 4 Kcal/mol. T-WHAM used to resolve barrier region
Distal
Proximal ligand-locked
Free Energy Landscape
Proximal ligand-free
Phe
87 2
Phe
87 1
1-Fe Distance [Å]
Population of proximal conformations
Conclusions
• REMD shows the conformational transition and supports thermal activation hypothesis
• Proximal state stabilized by conformational entropy
• Conformational states exist at all temperatures: relative populations change with temperature