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