Continuous Constant ph Molecular Dynamics

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1 Continuous Constant ph Molecular Dynamics Ana Patricia Gámiz-Hernández Department of Chemistry University of Basel AG Meuwly apgamiz (UniBas) CPHMD July / 46

2 Outline 1 Motivation Biopolymers in Solution Aminoacids and pka 2 Methods to compute pka values 3 MMTSB Tools CPHMD method Example: RNase A (PDB ID: 1RNU) apgamiz (UniBas) CPHMD July / 46

3 Outline Motivation Biopolymers in Solution 1 Motivation Biopolymers in Solution Aminoacids and pka 2 Methods to compute pka values 3 MMTSB Tools CPHMD method Example: RNase A (PDB ID: 1RNU) apgamiz (UniBas) CPHMD July / 46

4 Motivation Biopolymers in Solution Motivation (see Brooks papers) The stability and function of proteins are dependent on the environmental ph ph-dependent biological processes: ATP synthesis Enzyme catalysis membrane fusion protein folding apgamiz (UniBas) CPHMD July / 46

5 Motivation Biopolymers in Solution apgamiz (UniBas) CPHMD July / 46

6 Motivation Biopolymers in Solution [Khandogin and Brooks, PNAS 2007] apgamiz (UniBas) CPHMD July / 46

7 Motivation Biopolymers in Solution Motivation (see Brooks papers) The stability and function of proteins are dependent on the environmental ph In conventional MD the protonation state of the protein is preset and kept fixed according to the known pka (model compound) Two problems: the pka value of an ionizable side chain is typically unknown in its protein environment and the protonation/deprotonation of an ionizable side chain is a process that is coupled to the dynamics of the local environment (ionization equilibrium is neglected). apgamiz (UniBas) CPHMD July / 46

8 Motivation Biopolymers in Solution Q-HOP MD by Wei Gu A proton dissociates from acetic acid, diffuses in the water box and rebinds to the acetic acid. <Click> apgamiz (UniBas) CPHMD July / 46

9 Motivation Biopolymers in Solution Polyelectrolytes in Solution See D. Marx and Parinello papers Protonated water Enzyme Catalysis [JCTC 2013] [Science 2003] apgamiz (UniBas) CPHMD July / 46

10 Outline Motivation Aminoacids and pka 1 Motivation Biopolymers in Solution Aminoacids and pka 2 Methods to compute pka values 3 MMTSB Tools CPHMD method Example: RNase A (PDB ID: 1RNU) apgamiz (UniBas) CPHMD July / 46

11 Motivation Aminoacids and pka Titratable Amino acids ARG, LYS, HIS, CYS, TYR, GLU, ASP, CTER and NTER An amino acid (electrolyte) is a compound that ionizes when dissolved in ionizing solvents such as water. Side chain pka value (and backbone). CHARMM considers as initial guess the standard values at ph 7. the pka value of one amino acids inside the protein can change from standar values. apgamiz (UniBas) CPHMD July / 46

12 Methods to compute pka values Software: Static methods AccelrysPKA Accelrys CHARMm based pka calculation H++ Poisson-Boltzmann based pka calculations MCCE Multi-Conformation Continuum Electrostatics Karlsberg+ pka computation with multiple ph adapted conformations PETIT Proton and Electron TITration GMCT Generalized Monte Carlo Titration pkd server pka calculations and pka value re-design PROPKA Empirical calculation of pka values <Wikipedia> apgamiz (UniBas) CPHMD July / 46

13 Methods to compute pka values pk a values Generalities HA A + H + ph = log[h + ] and pk a = logk a = log [A ][H + ] [HA] ph = pk a + log [A ] [HA] G o prot = RTln10(pK a) apgamiz (UniBas) CPHMD July / 46

14 Methods to compute pka values References for CHARMM <CPHMD> CPHMD Tutorial M. S. Lee, F. R. Salsbury, Jr., and C. L. Brooks, III, Proteins, J. Khandogin and C. L. Brooks, III, Biophys. J J. Khandogin and C. L. Brooks, III, Biochemistry apgamiz (UniBas) CPHMD July / 46

15 Methods to compute pka values Other References and/or Programs <Gromacs> Dissociative Water Potential for Molecular Dynamics Simulations (JPC B, 2007, 111 (30), pp ) Specific Reaction Path Hamiltonian for Proton Transfer in Water: Reparameterized Semiempirical Models (JCTC, 2013, 9 (6), pp ) apgamiz (UniBas) CPHMD July / 46

16 Outline MMTSB Tools CPHMD method 1 Motivation Biopolymers in Solution Aminoacids and pka 2 Methods to compute pka values 3 MMTSB Tools CPHMD method Example: RNase A (PDB ID: 1RNU) apgamiz (UniBas) CPHMD July / 46

17 CPHMD method MMTSB Tools A set of Perl scripts Useful to set up the calculations of proteins: convpdb.pl conversion/manipulation of PDB files enercharmm.pl evaluate energy with CHARMM Black box: read/edit/modify/adapt the scripts. Topology and Parameter files just for standard amino acids. apgamiz (UniBas) CPHMD July / 46

18 A set of Perl scripts MMTSB Tools CPHMD method <MMTSB> apgamiz (UniBas) CPHMD July / 46

19 A set of Perl scripts MMTSB Tools CPHMD method <MMTSB> apgamiz (UniBas) CPHMD July / 46

20 A set of Perl scripts MMTSB Tools CPHMD method <MMTSB> apgamiz (UniBas) CPHMD July / 46

21 CPHMD method Setting up the System CHARMM set up CPHMD simulations using the single-temperature or replica-exchange (REX) protocol; analyze the titration data and compute pka s; obtain parameters for the potential of mean force (PMF) of model compound titration. apgamiz (UniBas) CPHMD July / 46

22 CPHMD method Tradicional molecular dynamics Protonation states of titratable residues assigned by comparing model pka s with the simulation ph kept fixed during simulation not realistic because pka s in the protein can be different than model pka s coexistence of both protonated and deprotonated states is neglected At ph 7: Glu deprotonated, His singly deprotonated, Lys and Arg protonated Initial guess apgamiz (UniBas) CPHMD July / 46

23 CPHMD method Constant ph molecular dynamics (PHMD) Basic idea Let solution ph dictate protonation states Combine sampling of conformational and protonation states Discrete PHMD Periodically stop MD and perform Monte-Carlo sampling of protonation states Continuous PHMD Molecular dynamics sampling of spatial and continuous titration coordinates Assumptions Source of protons not considered Kinetics of protonation not considered apgamiz (UniBas) CPHMD July / 46

24 CPHMD method Discrete PHMD: combine MD and MC sampling Baptista et al, J ChemPhys Monganet al, J ComputChem2004. apgamiz (UniBas) CPHMD July / 46

25 CPHMD method Pros and cons of discrete PHMD Conceptually simple and easy to implement. Solvent: PB or GB Proton charge on and off Discontinuous energy and force: instability Convergence for titration of multiple-sites may be slow Getting stuck in one prot. state: large conformational transition apgamiz (UniBas) CPHMD July / 46

26 Continuous PHMD (CPHMD) Lee, Salsbury Jr. Brooks III, Proteins 2004 CPHMD method Titration coordinates: λ i = sin 2 θ i Protonation states { 1 unprotonated; λ i = 0 protonated. (1) Propagation of titration coordinates f i = m i d 2 θ i dt 2 (2) H extend ({r i, θ i }) = H hybrid ({r i, θ i }) + i m i 2 θ i 2 + U ({θ i }) (3) apgamiz (UniBas) CPHMD July / 46

27 CPHMD method Coupling of titration and conformational dynamics Hybrid Hamiltonian H hybrid = U coulomb ({r i, θ i }) + U vdw ({r i, θ i }) + U GB ({r i, θ i }) (4) Linear interpolation for charge states q i,α = λ i q i,α unprot + (1 λ i )q i,α prot (5) Linear interpolation for van der Waals interactions { U vdw (1 λi )U = vdw (i, j) i=titrating residue; (1 λ i )(1 λ j )U vdw (i, j) i, j= both titrating residue. (6) apgamiz (UniBas) CPHMD July / 46

28 Biasing Potentials MMTSB Tools CPHMD method Biasing potential U bias = U model + U ph U barrier (7) PMF along the titration coordinate for the model compound ph dependence of protonation free energy U model = A(λ i B) 2 (8) U ph = 2.3RT λ i (pk model a )(i) ph (9) Harmonic potential for suppressing the population of mixed states, 0.1<λ i <0.9 U barrier = 4β i [ λ i 1 2] 2 (10) apgamiz (UniBas) CPHMD July / 46

29 CPHMD: double-site titrations CPHMD method apgamiz (UniBas) CPHMD July / 46

30 CPHMD: double-site titration CPHMD method apgamiz (UniBas) CPHMD July / 46

31 Outline MMTSB Tools Example: RNase A (PDB ID: 1RNU) 1 Motivation Biopolymers in Solution Aminoacids and pka 2 Methods to compute pka values 3 MMTSB Tools CPHMD method Example: RNase A (PDB ID: 1RNU) apgamiz (UniBas) CPHMD July / 46

32 Example: RNase A (PDB ID: 1RNU) RNase A: C-peptide from ribonuclease A Preparation of the Structure: HIS -> HSP add dummy hydrogen atoms to the carboxyl residues (Asp and Glu) -> ASSP2, GLUP2 N-terminal (NH2) and C-terminal (ACE) convpdb.pl -sel 1:13 -out charmm22 -segnames 1RNU.pdb \ sed "s/hsd/hsp/g" > cpept.pdb charmm pdbname=cpept < add_h_min.inp > add_h_min.out apgamiz (UniBas) CPHMD July / 46

33 Example: RNase A (PDB ID: 1RNU) Single-temperature CPHMD simulations we prepare a CHARMM input script for the constant ph Langevin simulation with the GBSW implicit-solvent model at ph 4. CPHMD is currently not implemented with explicit solvent. Output: /data/*.res, *.dcd, *.eng, *.pdb, and *.lambda charmm < cphmd-cpept.inp > cphmd-cpept.out phmd par 23 wri 25 npri 250 barr bartau 2.5 lam! par Unit number for PHMD parameter file (input) MUST specify.! ph Titration ph (default: 1.0)! barr Quadratic barrier height for each theta variable (def: 2.0! bartau Quadratic barrier height for each x variable (def: 2.5) apgamiz (UniBas) CPHMD July / 46

34 Example: RNase A (PDB ID: 1RNU) Replica-exchange CPHMD simulations CPHMD: convergence problem -> Large random errors in sampling protonation states -> Poor fitting to the Henderson-Hasselbachcurve In the replica-exchange method, simulations realize a random walk in the temperature space and do not get trapped in free-energy local-minimum states. REX-CPHMD is the recommended mode for running CPHMD simulations; improves the convergence of the titration property. aarex.pl -mdpar param=22x,xpar=par\_all22\_prot\_cmap\_phmd.inp, xtop=top\_all22\_prot_cmap\_phmd.inp, phmdpar=phmd.in,phmdpri=500,phmdph=4.0 apgamiz (UniBas) CPHMD July / 46

35 Example: RNase A (PDB ID: 1RNU) Process lambda files and calculate pka s From the above REX-CPHMD simulations, you will obtain one lambda file in each replica directory that looks like this. # ititr ! Running number # ires ! Residue number frompdb # itauto ! 0 -single 3,4- COO, 1, ! step number, lambda and x values lambda value for Lys lambda value for Glu x value for Glu lambda value for Lys-7 apgamiz (UniBas) CPHMD July / 46

36 Example: RNase A (PDB ID: 1RNU) CPHMD: enhance conformational sampling Independent replicas running at different temperature Periodic swap both conformational and protonation states Accelerate crossing of local energy barriers extractlamb.pl [# replicas] [# rex steps] [lambda printing frq] [# apgamiz (UniBas) CPHMD July / 46

37 Example: RNase A (PDB ID: 1RNU) Process lambda files and calculate pka s Obtain λ values and unprotonated fractions from REX-CPHMD simulations Calculate pka values by fitting to Henderson-Hasselbach (HH) equation. cptpka.pl cpept ph-3.0.ens sx ph-4.0.ens sx!output cpept.pka (resnumber, pka, Hill coefficient, fitting correlation c cpept-res-*.ps (titration plots for titrating residues) cpept-res-*.dat (S values for titrating residues) cpept-res-*.fit (fitting output from xmgrace) apgamiz (UniBas) CPHMD July / 46

38 Example: RNase A (PDB ID: 1RNU) CPHMD: how to compute pka values Compute pka by fitting unprotonated fractions, S, at different ph, to the generalized Henderson-Hasselbach (Hill) equation: S = ρ unprot ρ unprot + ρ prot = n(pka ph) (11) apgamiz (UniBas) CPHMD July / 46

39 Example: RNase A (PDB ID: 1RNU) CPHMD: how to compute pka values S = n(pka ph) (12) apgamiz (UniBas) CPHMD July / 46

40 Example: RNase A (PDB ID: 1RNU) Pros and cons of CPHMD Continuous energy and force May converge faster than DPHMD for multi-site titrations Accuracy can be systematically improved Well suited for studying ph-coupled conformational dynamics Has been validated using both pka calculations and ph-dependent conformational dynamics Not easy to implement Has unphysical states (intermediate lambda values): but populations can be suppressed to be low Non-trivial to incorporate explicit solvent apgamiz (UniBas) CPHMD July / 46

41 Example: RNase A (PDB ID: 1RNU) Derive parameters for model PMF functions Parameters for the model PMF of Lys, Asp, His U mod = R 1 sin 4 θ +R 2 sin 2 θ +R 3 [sin 2 (θx) R 4 ] 2 +R 5 sin 4 θ +R 6 sin 2 θ apgamiz (UniBas) CPHMD July / 46

42 Erros in CPHMD MMTSB Tools Example: RNase A (PDB ID: 1RNU) The accuracy of CPHMD is around 1 pk unit, althought the discrepancy is larger for buried residues. Marjor sources of errors: implicit solvent model (GB) Van der Waals surface used in GB underestimates Born radii Too much self solvation: G = G selfsolvation + G screeningcoumb Other potential errors: Larger errors for Lys and Arg -> multi-site titration Force field bias: for small peptides and less for native-state simulations Polarization effects neglected apgamiz (UniBas) CPHMD July / 46