Orthogonal Space Sampling of Slow Environment Responses

Transcription

1 IMA University of Minnesota 2015 Orthogonal Space Sampling of Slow Environment Responses Lianqing Zheng, Chao Lv, Dongsheng Wu, William Harris, Xubin Li, Erick Aitchison, and Wei Yang Institute of Molecular Biophysics & Department of Chemistry and Biochemistry Florida State University 1

2 Free Energy Sampling: Following Perturbation along Order Parameters to Sample Important Events in the Orthogonal Spaces Geometric Perturbation (PMF Calculations) Chemical Perturbation (FEP Calculations) Transition State Zeroth-Order Importance Sampling Methods: -- Umbrella Sampling (Restraining)/ Thermodynamic Integration (Constraining) -- Non-equilibrium regime umbrella sampling: Steered molecular dynamics; Non-equilibrium regime TI: fast growth TI ΔG An Order Parameter (for instance: Distance) 2

3 Environment Responses Freedom Reduction Catastrophe in Zeroth-Order Importance Sampling Methods (US or TI) II III B A I I. Traverse effect renders US challenging to work; II.US hides problems: Order parameter distribution overlap checking does not tell us anything about whether collected samples from two neighboring states overlap in the orthogonal space (along the degrees of freedom orthogonal to the order 3 parameter).

4 Ergodic Sampling Checking and Order Parameter Space Random Walk (First-Order Generalized Ensemble Sampling) U ( ) U o fm( ) fm( ) G( ) I. A Necessary Condition for Ergodic Sampling Checking: When a system leaves state A towards state B, it can be propagated back within finite time and vice verse. II.Order Parameter Space Random Walk: To preserve dynamic degrees of freedom while crossing free energy barriers along OPs. III.Two General Strategies: (a) Recursive sampling to adaptively update OP-dependent biasing potentials; (b) Umbrella Sampling Hamiltonian Replica Exchange. 4

5 Recursion (Adaptive Sampling) Methods to Enlarge Fluctuations in the Order Parameter Space U( ) U o fm( ) fm( ) G( ) I. Adaptive umbrella sampling method: slow recursion; robust and convergent (long-tail); easy to implement i t Gt ( ) RT ln( ) fm ( ) i 1 i II. Metadynamics: fast recursion; lack of robustness and oscillatory; easy to implement (t ) 2 i G( ) hexp 2 2w t III. Adaptive biasing force: fast recursion; robust and convergent (short-tail); can be challenging to implement ln J G( ) F ' d ' F U o RT i o 5

6 The Hidden Barrier (Orthogonal Space Sampling) Issue: How to Sample Slow Environment Responses Z A + - TS Nonpolar Moiety Polar Moiety B Hidden responses Slow processes are likely to be collective; activating cooperative energy fluctuations is remarkably challenging. 6

7 The Elephant Has Been in the Room for a While -- Gated motions -- Environment mediated dynamics -- Enslaved dynamics -- Hamiltonian lagging issue Slow responses in the orthogonal space need to be activated; Within the generalized ensemble framework, response OP (ROP) needs to be predefined in a system-independent manner. Hidden responses If hidden processes are activated events, ROPs cannot be learned from lower-level sampling results. 7

8 How to Define ROP: Generalization of the Marcus Theory The Marcus Theory OP: Vertical Energy Gap -- Strong-Coupled ROP (SC-ROP) With U( ) U o fm( ), the whole process can be considered as a series of Generalized Electron Transfer sub-processes. + ln J Uo F RT The Key Generalization: for any target process, generalized force is a generic SC-ROP for the dynamics along. Zheng, L., Chen, M., and Yang, W. J. Chem. Phys. 130, (2009). 8

9 Orthogonal Space Random Walk: the Beginning of the Second-Order Generalized Ensemble Sampling Scheme First-order generalized ensemble: random walk in the order parameter space U( ) U o fm( ) fm( ) G( ) Orthogonal space random walk: random walks in both the target OP and the SC-ROP directions. U( ) U o fm( ) gm(, F ) fm( ) Go ( ) gm[, F ] Go' (, F ) ensemble under [U o Go ( )] Note: (a) No system-dependent orthogonal collective variable needs to be preselected to accelerate sampling of slow environment responses; (b) Strongly-coupled responses: only involving direct interactions between the perturbation center and its environment, but not environment internal interactions. Zheng, L., Chen, M., and Yang, W. J. Chem. Phys. 130, (2009). 9

10 Generalized Force as the Order Parameter to Explore the Orthogonal Space F Energy Surface Hidden responses TS ln J Uo RT TS The General Formula of the Orthogonal T T U m U 0 fm ES 0 gm, F Space Sampling Scheme: TES (TES-T0)/TES : scaling factor to confine sampling exploration in the orthogonal space Zheng, L.; Chen, M.; Yang, W. PNAS USA 2008, 105, ; Zheng, L.; Chen, M.; Yang, W. J. Chem. Phys. 2009, 130, ; Zheng, L.; Yang, W. J. Chem. Theor. Comput. 2012, 8,

11 Alchemical Free Energy (Free Energy Perturbation) Simulation Binding Affinity Change; Solvation Energy; pka Shift; Redox Potential; Electron Transfer Potential etc. + A Gbinding(A) GSolutionA->B + A GProteinA->B Gbinding(B) B B G = Gbinding(B) - Gbinding(A) = GProteinA-> B - GSolutionA->B 11

12 -Dynamics Allows the Unification of PMF and FEP Sampling The Extended Hamiltonian: a λ- dynamics generalization 0 λ 1 U o ( ) (1 )U sa U sb U e p 2 H Ho 2m Um Uo( ) fm( ) TES To gm,f TES Kong, X.; Brooks, C. L. J. Chem. Phys. 1996, 105, Zheng, L.; Yang, W. J. Chem. Theor. Comput. 2012, 8,

13 FMN reduction potential calculation (QM/MM: SCCDFTB/CHARMM) du/dλ kcal/mol λ λ Free energy kcal/mol du/dλ kcal/mol What to Play with as a Beginner User 1.Checking traveling; monitoring F fluctuations; identifying sampling bottleneck regions; 2.Checking gm(, F ) and the free energy derivative curve dg/d ; 3.Monitoring free energy convergence; 4.Analyzing slow hidden events responsible for detected sampling bottlenecks. 13

14 H2S versus H2O Trans-Membrane Permeations Lv, C. et al. J. Comput. Chem. in press (2015) 14

15 High-Order Orthogonal Space Tempering (HOOST) U m U o ( ) fm( ) ggm, F hhm, F, FF F U o ( ) 1. Essential weakly-coupled environment responses are defined as Responses to F Fluctuations ; 2. The order parameter: ln J F U o ( ) FF RT F F 3. It involves environment internal interactions; 4. h is also a parameter defining sampling aggressiveness and sampling boundary; 5. Accelerating weakly-coupled environment responses can be essential to wetting/de-wetting transitions and long-range conformational responses etc. 15

16 For instance: Reduction Potential of FMN inside a Protein gm, F hm 0.5, F,FF Essential Long Range Conformational Transitions are Selectively Coupled. 16

17 pka Prediction: Lys66 of Δ+PHS Form of Staphylococcal Nuclease Both long-range and short-range motions need to be activated. Free energy results: protein: kcal/mol solution: kcal/mol G: 7.71 kcal/mol pka shift: 5.7 Expt. pka shift: 5.2 ( ) 17

18 pka Prediction: Lys66 of Δ+PHS Form of Staphylococcal Nuclease 18

19 pka Prediction: Lys66 of Δ+PHS Form of Staphylococcal Nuclease 19

20 Hydration Free Energy of a Linear Alkane C16H34 SOLV C16H34 (gas) C16H34 (aqueous) gm(, F ) At the dummy state, all external interactions are turned off while all internal interactions are kept. To exam which aspect sampling is more crucial: torsional sampling or water environment sampling 20

21 Hydration Free Energy of a Linear Alkane C16H34 DOCK C16H34 (gas) C16H34 (aqueous) gm(, F ) C16H34 (dummy) DOCK At the dummy state, all the electrostatic terms and the selected torsional terms are turned off. To exam which sampling is more crucial: torsional sampling or water environment sampling 21

22 Hydration Free Energy of a Linear Alkane C16H34 C16H34 (gas) SOLV C16H34 (aqueous) C16H34 (dummy) DOCK SOLV: At the dummy state, all external interactions are turned off while all internal interactions are kept. DOCK: At the dummy state, all the electrostatic terms and the selected torsional terms are turned off. To exam which sampling is more crucial: torsional sampling or water environment sampling 1.From SOLV: Ghyd : 6.33 kcal/mol From DOCK: Ghyd: 6.28 kcal/mol ( ) LRC (long-range correction): kcal/mol Overall: 4.87 kcal/mol 2.The experimental value: 5.0 kcal/mol 3.Water environment response sampling is more crucial. It enslaves the folding and unfolding of the alkane chain. 22

23 SAMPL4 Host-Guest Challenge: Cucurbit[7]uril (CB7) to Test A Preliminary High-Order Orthogonal Space Tempering 1.Fully automated. 2.At the dummy state, the center atom is restrained within a sphere (simultaneously searching binding poses); and all the torsional and electrostatic terms are turned off ns MD simulation for each ligand binding prediction; 3.Compensating ion appearance is coupled with ligand annihilation; 4.AMBER-GAFF force field (TIP3P). 23

24 SAMPL4 Host-Guest Challenge: Cucurbit[7]uril (CB7) to Test A Preliminary High-Order Orthogonal Space Tempering 24

25 SAMPL4 Host-Guest Challenge: Cucurbit[7]uril (CB7) to Test A Preliminary High-Order Orthogonal Space Tempering Kendall Rank 25

26 Analyzing a Host-Guest Binding Process 26

27 Analyzing a Host-Guest Binding Process 27

28 Hidden Events in the Host-Guest Binding Process Number of waters Angle d (Å) d (Å) Cirularity d (Å) 28

29 GPU-Enabled Applications on Large Biomolecules DNA repair enzyme base flipping Triose-phosphate isomerase (TIM) 29

30 Predictive Sampling Based on Potential Scaling Perturbation A General Expanded Hamiltonian: A Generalized Orthogonal Space Tempering (gost): Although varying λ may lead to a shifted distribution, it can take very long time for a configuration to dynamically evolve to new important regions; i.e. slow configuration responses, particularly ones involving cooperative fluctuations, are likely to be the major sampling bottleneck. 30

31 Deca-Alanine Peptide Folding and Refolding It is especially challenging for CHARMM22-CMAP; so it becomes a better model system. Lambda [0.81, 1.001] Native N-2 31

32 Deca-Alanine Peptide Folding and Refolding Water structural collective fluctuations need to be activated. The end-to-end distance is horrible The C-terminus is over stabilized in comparison with the N-terminus.

33 How About Other Methods? Total Potential Energy: Three common SE(lambda) functions: Original; Solute Tempering, Solute Tempering 2 33

34 Accurate Prediction of Protein Functional Dynamics: Population Shift Model Verification Adenylate Kinase, No Ligand, Starting from the apo Conformation RMSD ~ 1.3 Angstroms 34

35 Acknowledgment Group Members Prof. Wei Yang (PI) Dr. Dongsheng Wu (Postdoc) William Harris Erick Aitchison Xubin Li Dr. Chao Lv* (currently at WUSTL) Dr. Donghong Min* Dr. Mengen Chen* * Former members Computing Support FSU Research Computing Center (RCC); FSU IMB Computing Facility; Oak Ridge Super-Computing Center (TITAN), Texas Austin Computing Center (TACC); Argonne National Laboratory Super-Computing Center Funding Support 35