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### Network models of Replica Exchange

Exploring landscapes for protein folding and binding using replica exchange simulations

Effective potential

Important coordinates

- 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)

- 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 Born

Surface area model

Born radius-based estimator

E. Gallicchio, and R.M. Levy, JCC, 25, 479 (2004)

AGBNP: Pairwise Descreening Scheme

Born radii: rescaled pairwise descreening approximation:

j

i

Rescale according to self-volume of j:

Self-volume of j (Poincarè formula, ca. 1880):

Hawkins, Cramer, and Truhlar, JPC 1996

Schaefer and Karplus, JPC 1996

Qiu, Shenkin, Hollinger, and Still, JPC 1997

E. Gallicchio, R. Levy, J. Comp. Chem. (2004)

Non-Polar Hydration Free Energy

Non-polar hydration free energy estimator:

: Surface area of atom i

: Estimator based on Born radius

: Surface tension and van der Waals adjustable parameters

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: 514

Zuckerman & Lyman (2006) JCTC 2: 1200 (with erratum)

Replica exchange molecular dynamics

rough energy landscapes and distributed computing

MD

MD

MD

MD

MD

700 K

450 K

320 K

energy

Y. Sugita, Y. Okamoto

Chem. Phys. Let., 314,

261 (1999)

“important coordinates”

walker 1

walker 2

energy

walker 3

walker 4

200 K replica

“important coordinates”

Replica exchange molecular dynamics

rough energy landscapes and distributed computing

MD

MD

MD

MD

MD

MD

MD

700 K

450 K

replica

320 K

200 K

Y. Sugita, Y. Okamoto (1999) Chem. Phys. Let., 314:261

U2

F1

U1

Protein folding: REM and kinetic network models- free energy surfaces of the GB1 peptide from
- REM and comparison with experiment

- kinetic network model of folding pathways
- for GB1

Andrec M, Felts AK, Gallicchio E, Levy RM.. PNAS (2005) 102:6801.

- kinetic network model of REMD
- (simulations of simulations)

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 b-Hairpin of B1 Domain of Protein G

Simple (surf area) nonpolar model

OPLS/AGBNP

-hairpin > 90%

-helix < 10%

G ~ 2 kcal/mol

The potential of mean force of the capped peptide.

A Felts, Y. Harano, E. Gallicchio, and R. Levy, Proteins, 56, 310 (2004)

Kinetic network models for folding

Andrec, Felts, Gallicchio & Levy (2005) PNAS, 102, 6801

Network nodes are snapshots from multiple temperatures of a replica exchange simulation.

Tcold

Thot

800,000 nodes

7.4 billion edges

Dynamical/kinetic considerations:

Transition rates (edges) are motivated by Kramers theory: transitions are allowed if there is sufficient structural similarity, and forbidden otherwise.

Simulations are performed using the Gillespie algorithm for simulating Markov processes on discrete states:

- Waiting time in a state is an exponential random variable with mean = 1/(Sj kij)
- Next state is chosen with probability proportional to kij

Equilibrium considerations:

Sufficiently long trajectories must reproduce WHAM results.

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 j from temperature TB having energy Ej

node i from temperature TAhaving energy Ei

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

b

a

t = 9 units ≈ 180 ns

t = 2500 units ≈ 50 µs

b

a

Fraction of hairpin conformation averaged over 4000 stochastic trajectories run at 300 K and begun from an initial state ensemble equilibrated at 700 K.

91% of 4000 temperature-quenched stochastic trajectories begun from high-energy coil states pass through states with a-helical content

Andrec, Felts, Gallicchio & Levy (2005) PNAS, 102, 6801

Evidence for a-helical intermediates in b-sheet folding and misfolding

- Non-native helices have been observed in b-lactoglobulin folding

- Rapid formation of a structure
- Can exist as a stable thermodynamic species and as intermediates
- May be important in protecting exposed ends of b-sheet from intermolecular interactions

Forge, Hoshino, Kuwata, Arai, Kuwajima, Batt & Goto (2000) JMB 296:1039

Kuwata, Shastry, Cheng, Hoshino, Batt, Goto & Roder (2001) Nat. Struct. Biol. 8:151

- Amyloid b-sheets can form from a-helical precursors

- Myoglobin and coiled-coil proteins can form amyloid fibrils

Fändrich, Forge, Buder, Kittler, Dobson & Diekmann (2003) PNAS 100:15463

Kammerer, Dobson, Steinmetz et al. (2004) PNAS 101: 4435

- Fibril formation in amyloid b-protein may occur via a helical intermediate

Kirkitadze, Condron & Teplow (2001) JMB 312:1103

Fezoui & Teplow (2002) JBC 277: 36948

- Computational and theoretical evidence

- Helical structures have been observed in G-peptide simulations

García & Sanbonmatsu (2001) Proteins 42:345

Zagrovic, Sorin & Pande (2001) JMB 313:151

Wei, Mousseau & Derreumaux (2004) Proteins 56:464

- Entropy-stabilized helical intermediates may be generic in b-sheet protein folding landscapes

Chikenji & Kikuchi (2000) PNAS 97:14273

Exploring landscapes for protein folding and binding using replica exchange simulations

Effective potential

Important coordinates

- 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

U2

F1

U1

F2U1

U2U1

U2F1

F2F1

F1U2

U1U2

F1F2

U1F2

One walker

Two walkers

N walkers

ku2

ku

kuN

F

U

kf2

FN

UN

kRE

kRE

kf

kfN

ku1

kf1

ku and kf: physical kinetics

kRE: replica exchange “kinetics”

ku2

F2

U2

kf2

kRE

ku1

F1

U1

kf1

5 walkers: 3840 states

N walkers: 2N N! states

2 walkers: 8 states

Convergence at low temperature depends on the number of F1 to U1 to F1 “transition events”

Gillespie “simulation of protein folding simulations”

†

Speed limit for replica exchange efficiencyThe 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

Zheng W, Andrec M, Gallicchio E, Levy RM. PNAS (2007) 104:15340.

- 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 on a 2-d continuous potential with an entropic barrier to folding

2-d continuous potential

Potential energy along x

F

U

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

Reverse-engineering rates

fex= 10-4

fex= 10-2

kf(T1)

6.1

6.4

6.3

ku(T1)

0.0036

0.0038

0.0037

kf(T2)

0.30

0.29

0.31

ku(T2)

0.42

0.42

0.43

Reverse-Engineering rates from the trajectory on the continuous potential using lifetime & branching ratios

T1=296K

T2=474K

RE on the continuous potential vs RE on the kinetic network

The faster the replica exchange rate, the bigger the discrepancy.

Kinetic network

Continuous potl

Total # of transitions

fex = 5·10-2

fex = 5·10-3

fex = 1·10-3

Infinitely fast exchange limit*

*Calculated using harmonic mean of rate constants

Non-Markovian effects -- History dependence

Probability

Calculated from the continuous traj. at different exchange rates

fex =5·10-3

fex =10-4

P(U2F1U1F2)

0.849

0.103

P(F2F1U2F1U1F2)

0.521

0.094

P(U2F1F2F1)

0.150

0.886

P(F2F1U2F1F2F1)

0.477

0.895

- 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.

Exploring landscapes for protein folding and binding using replica exchange simulations

Effective potential

Important coordinates

- 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

Protein Folding with All Atom Potentials

Insights using Replica Exchange and Network Models

Effective potential

Important coordinates

- 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

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

Reduced Coordinate

Folding Landscape

Binding Landscape

The P450 puzzle

- Cytochrome P450s metabolize many aliphatic molecules and 90% of pharmaceutical ligands

NPG

P450BM-3/NPG

Phe87

Heme

- Several X-ray crystal structures of P450s show substrate distant from active site
- Hypothesized a conformational equilibrium between productive and unproductive conformational states

Experimental and Modeling Clues

- UV-VIS and SSNMR experiments indicate temperature-dependent equilibrium between Fe-bound and un-bound species.

Induced Fit docking finds a Fe-bound conformation of higher energy than the X-ray conformation

ω1-Fe

X-ray (Distal)

Induced Fit Model* (Proximal)

- 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?

*Jovanovic, T.; Farid, R.; Friesner, R. A.; McDermott, A. E. J. Am. Chem. Soc. 2005, 127, 13548.

REMD of P450 NPG Complex

- Provides populations of conformational states (canonical sampling) as a function of temperature
- Can be used to construct free energy landscape

- Model ligand and 120 active site residues
- 24 replicas between 260 and 463 K
- 72 ns total aggregate simulation time

Ravindranathan, K.P., E. Gallicchio, R.A. Friesner, A.E. McDermott, and R.M. Levy.

J. Am. Chem. Soc.,128, 5786-5791 (2006).

Population of proximal conformations

Free Energy LandscapePhe87 1

Proximal ligand-free

- 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

Phe87 2

Proximal ligand-locked

1-Fe Distance [Å]

Distal

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

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