1. Exploring landscapes... “important coordinates” energy 700 K replica 200 K replica

2. Important coordinates Effective potential 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

3. 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 modelBorn radius-based estimator E. Gallicchio, and R.M. Levy, JCC, 25, 479 (2004)

4. AGBNP: Pairwise Descreening Scheme i Born radii: rescaled pairwise descreening approximation: Rescale according to self-volume of j: Self-volume of j (Poincarè formula, ca. 1880): E. Gallicchio, R. Levy, J. Comp. Chem. (2004) Hawkins, Cramer, and Truhlar, JPC 1996 Schaefer and Karplus, JPC 1996 Qiu, Shenkin, Hollinger, and Still, JPC 1997 j

5. 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)

6. 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)

7. 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 & Garc í a (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)

8. Replica exchange molecular dynamics rough energy landscapes and distributed computing 200 K MD 700 K “important coordinates” energy 450 K 320 K Y. Sugita, Y. Okamoto Chem. Phys. Let., 314, 261 (1999)

9. Replica exchange molecular dynamics rough energy landscapes and distributed computing 200 K MD 700 K 450 K 320 K Y. Sugita, Y. Okamoto (1999) Chem. Phys. Let., 314:261 “important coordinates” energy 700 K replica 200 K replica walker 4 walker 1 replica MD walker 2 walker 3

10. 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) F2F2 U2U2 F1F1 U1U1 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.

11. 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

12. The  -Hairpin of B1 Domain of Protein G The potential of mean force of the capped peptide. Simple (surf area) nonpolar modelOPLS/AGBNP A Felts, Y. Harano, E. Gallicchio, and R. Levy, Proteins, 56, 310 (2004)  -hairpin > 90%  -helix < 10%  G ~ 2 kcal/mol

13. 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 k ij ) 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 nodes 7.4 billion edges T cold T hot Andrec, Felts, Gallicchio & Levy (2005) PNAS, 102, 6801

14. Connection between kinetic model and equilibrium populations Equilibrium populations for temperature T 0 are preserved if for each pair of nodes (i, j) the ratio of transition rates follows WHAM weighting: node i from temperature T A having energy E i node j from temperature T B having energy E j where f A (  0 ) and f B (  0 ) are free energy weights for the T A and T B simulations at reference temperature T 0 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

15. 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

16. 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 312:1103 Fezoui & Teplow (2002) JBC 277: 36948 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  -sheets can form from  -helical precursors Fändrich, Forge, Buder, Kittler, Dobson & Diekmann (2003) PNAS 100:15463 Kammerer, 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 42:345 Zagrovic, Sorin & Pande (2001) JMB 313:151 Wei, 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

17. Important coordinates Effective potential 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

18. Network models of Replica Exchange F U kuku kfkf F1U2F1U2 U1F2U1F2 U1U2U1U2 F1F2F1F2 F2U1F2U1 U2F1U2F1 U2U1U2U1 F2F1F2F1 One walker Two walkers F2F2 U2U2 F1F1 U1U1 2 walkers: 8 states N walkers F2F2 U2U2 F1F1 U1U1 FNFN UNUN 5 walkers: 3840 states N walkers: 2 N N! states Gillespie “simulation of protein folding simulations” k RE k u2 k f2 k u1 k f1 k RE k u1 k f1 k u2 k f2 k uN k fN k u and k f : physical kinetics k RE : replica exchange “kinetics” Convergence at low temperature depends on the number of F 1 to U 1 to F 1 “transition events”

19. Speed limit for replica exchange efficiency † † Temperature of high- temperature replica T 2 (K) Number of transition events at low temperature T 1 in 1 ms Arrhenius case (∆C p † = 0) Non-Arrhenius case (∆C p † < 0) 300111 4401049532 7002801134 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 (∆C p † < 0)

20. 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.

21. 2-d continuous potential Potential energy along x FU 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

22. Rate constants extracted from (Uncoupled) simulations on the 2-d continuous potential kuku kfkf

23. f ex = 10 -2 f ex = 10 -4 0.430.42 k u (T 2 ) 0.310.290.30 k f (T 2 ) 0.00370.00380.0036 k u (T 1 ) 6.36.46.1 k f (T 1 ) Reverse-engineering ratesUncoupled Reverse-Engineering rates from the trajectory on the continuous potential using lifetime & branching ratios T 1 =296K T 2 =474K

24. 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 f ex = 5·10 -3 f ex = 1·10 -3 f ex = 5·10 -2 * Calculated using harmonic mean of rate constants Total # of transitions

25. f ex =5·10 -3 0.8860.150 P(U 2 F 1  F 2 F 1 ) 0.8950.477 P(F 2 F 1  U 2 F 1  F 2 F 1 ) 0.0940.521 P(F 2 F 1  U 2 F 1  U 1 F 2 ) 0.1030.849 P(U 2 F 1  U 1 F 2 ) Calculated from the continuous traj. at different exchange rates Probability Non-Markovian effects -- History dependence f ex =10 -4

26. 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

27. Important coordinates Effective potential 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

28. Important coordinates Effective potential 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

29. Potential of mean force (PMF) along x FU

30. 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

31. 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 LandscapeBinding Landscape Reduced Coordinate

32. 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

33. 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?

34. 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).

35. 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 Phe87  2 Phe87  1  1-Fe Distance [Å] Population of proximal conformations

36. 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