How much detail is sufficient? A comparison of non-polarizable and - - PowerPoint PPT Presentation

How much detail is sufficient?

A comparison of non-polarizable and polarizable force fields for protein folding

Anthony Hazel July 12, 2018 NAMD Workshop DICP , Dalian, CN

Free Energy Landscapes

• Protein folding landscapes are narrower at

the bottom; there are few low-energy, native- like conformations and many more open unfolded structures.

• Random steps that are mostly incrementally

downhill in energy

Wolynes, Onuchic, Thirumalai. Navigating the folding routes. Science 267: 1619-1620 (1995).

Unfolded States Local Structures Global Structure Native State

Free energy: G = H - TS

Umbrella Sampling

Kumar et al. J. Comput. Chem. 13:1011 (1992). Park et al. J. Chem. Phys. 119:3559 (2003).

Potential of Mean Force (PMF)

N independent simulations

k1(ξ-ξ1)2 k2(ξ-ξ2)2 kN-1(ξ-ξN-1)2 kN(ξ-ξN)2

WHAM Equations Unbiased Hamiltonian Bias Potential Arbitrary constant for each simulation Unbiased PMF Biased PMF Subtract bias Initial guess

• f constants (fm = 0)

… convergence of fm’s

Replica Exchange Molecular Dynamics (REMD)

Allow multiple, parallel simulations (replicas) to periodically exchange parameters (temperature, biasing potentials, etc.) Hamiltonian exchange for US simulations (REUS) Exchange probability given by Metropolis criterion:

P(i ↔ j) = min {1, exp(−β∆E)}

Sugita et al. J. Chem. Phys. 113:6042 (2000).

∆E = [Ei(qi) + Ej(qj)] − [Ej(qi) + Ei(qj)]

β = (kBT)−1

Replica Exchange Umbrella Sampling (REUS)

Park & Im. J. Chem. Theory Comput. 10:2719-2728 (2014).

Toy Model: U(x,y) Sampling Free Energy

P(y|x) US REUS aREUS

B1 domain of streptococcal protein G (GB1)

C-terminal hairpin of GB1 (G41-E56)

• A. All residues B. Hydrogen bonds C. Hydrophobic core

PDB: 1GB1 56 residues

GB1 as a model for β-sheet folding

Munoz et al. Nature. 149:072317 (1997). Fesinmeyer et al. J. Am. Chem. Soc. 126:7238-7243 (2004).

30-50% folded @ 298/300K Folds in ~6μs

Best and Mittal. Proteins. 79:1318-1328 (2011).

Commonly used to calibrate force fields and enhanced sampling techniques

CHARMM Drude Polarizable Force Field

Drude Particle Parent Atom SWM4-DNP Water Lone Pairs

Drude Lysine

Non-polarizable FFs tend to be

• verpolarized in order to mimic a

solvent environment Drude oscillator polarizable FF splits each heavy atom into a (+) parent atom and a (—) Drude particle, connected by a stiff spring (kD) Drude model allows for polarization of molecule from environment, not just molecular geometry drude on drudeTemp 1 drudeDamping 20.0 drudeBondLen 0.25 drudeHardWall on drudeNBTHOLEcut 5.0 LJcorrection yes

Lopes et al. J. Chem. Theory Comput. 9:5430−5449 (2013). Huang and MacKerell, Jr. Biophys. J. 107:991–997 (2014).

α = qD2/kD

Polarizability

1-fs timestep

Folding PMFs of the GB1 β-hairpin

89-100 REUS windows 12-20ns/window

ΔGfold = -7.2±2.2 ΔGfold = -1.7±2.2 ΔGfold = +2.8±1.7 ΔGfold (exp) = 0.0-0.5 kcal/mol C36 greatly overestimates ΔGfold C22* slightly overestimates ΔGfold Drude slightly underestimates ΔGfold

Side chain hydration free energies show room for improvement in the non-polarizable and polarizable models

The Drude model underestimates hydroxyl, overestimates some charged, and improves sulfur-containing and amide hydration free energies. Improved backbone hydration needs to be compensated by improved N-H—O=C hydrogen bonding or peptide will unfold. Also, while backbone (NMA) hydration is improved in the Drude model over C36, further improvements could still be implemented.

40-stage Weeks–Chandler– Anderson (WCA)- decomposition free-energy perturbation (FEP) procedure

• Y. Deng and B. Roux. J. Phys.
• Chem. B. 108:16567 (2004).
• Y. Deng and B. Roux. J. Phys.
• Chem. B. 113:2234 (2009).

Chris Rowley

Examining backbone polarization in the Drude model

Dipole moments are enhanced in the Drude model N-H bonds significantly polarize during intrapeptide hydrogen bonding, but not when hydrogen bonding with water C=O bonds behave in the opposite manner, only polarizing significantly when hydrogen bonding with water Unlike water-peptide hydrogen bonding, intrapeptide hydrogen bonding aligns parent-drude bond with chemical bond

C36 Protein

4 6 8 10 12 14 16

RG (Å) C22* Drude

30 60 90 120 150

Dipole Moment (D) Backbone

1 2 3 4 5 4 6 8 10 12 14 1 2 3 4 5

Nhb

6 1 2 3 4 5 6 12 18 24 30

− −

hb

− −

hb

N−DN HBonds

C

4 6 8 10 12 14

N−DN Other

0.4 0.5 0.6 0.7 0.8 0.9

C−DC HBonds

4 6 8 10 12 14

RG (Å) C−DC Other

0.2 0.3 0.4

Dipole Moment (D) O−DO HBonds

1 2 3 4 5

Nhb

4 6 8 10 12 14

O−DO Other

6 1 2 3 4 5 0.1 0.2 0.3

D

N−DN HBonds

4 6 8 10 12 14

N−DN Other

35 40 45 50 55 60

C−DC HBonds

4 6 8 10 12 14

RG (Å) C−DC Other

35 40 45 50 55 60

Angle (deg) O−DO HBonds

1 2 3 4 5

Nhb

4 6 8 10 12 14

O−DO Other

6 1 2 3 4 5 35 45 55 65 75 85

Altering backbone (and side chain) polarizabilities in the Drude model

By rescaling the histograms in WHAM, we can recalculate the PMF with different parameters using the states already sampled by our REUS simulations:

NMA

∆Ghydr

elec (kcal/mol)

α´

N*H/αN*H

−15 −10 1.0 1.5 2.0 NMA Ser Thr

∆Ghydr

elec (kcal/mol)

α´

O*H/αO*H

−15 −10 −5 1.0 1.5 2.0

A B

αN*H

A PMF (kcal/mol) Nhb

−5 5 10 15 20 25 1 2 3 4 5 αN*H

A

αO*H 6 1 2 3 4 5 0.0 0.5 1.0 1.5 2.0

α´

i*H/αi*H

αN*H

A

αO*H

B ∆Gfold (kcal/mol) α´

i*H/αi*H

Exptl αO*H αN*H −5 5 10 15 20 25 0.0 0.5 1.0 1.5 2.0 αN*H

A

αO*H

B ∆Gfold (kcal/mol) α´

i*H/αi*H

Exptl αO*H αN*H −5 5 10 15 20 25 0.0 0.5 1.0 1.5 2.0

histogram(rij) = P

t exp

• − β(Unew − Uold)
• δ(r(t) − rij)

P

t exp

• − β(Unew − Uold)
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Increasing the backbone N polarizabilities by 60% showed the best improvement for both the backbone hydration and GB1 β-sheet folding

Checking the altered polarizabilities with an α-helical peptide, Ala10

C36

0.0 0.2 0.4 0.6 0.8 12 16 20 24 28 32

End−to−end Distance (Å) C36 C22*

0.0 0.2 0.4 0.6 0.8

α−Helical Content C36 C22* Drude

1.0 0.0 0.2 0.4 0.6 0.8 2 4 6 8 10

PMF (kcal/mol)

The Drude model is already well

• ptimized for α-helices

Increasing in N polarizability by >30% would deteriorate quality of the model 30% increase is gives a good balance between α-helices and β- sheets

α

α−

−

α α

α

−

α−

− α

− −

− −

α

− −

B ∆Gfold (kcal/mol) α´

N*H/αN*H

Ala5 αN*H 5 10 0.0 0.5 1.0 1.5 2.0 α

− −

B ∆Gfold (kcal/mol) α´

N*H/αN*H

Ala5 αN*H 5 10 0.0 0.5 1.0 1.5 2.0 α

− −

B

− −

− −

Conclusions

We used replica exchange umbrella sampling (REUS) to calculate the folding free energies of the model β-sheet peptide, GB1 The polarizable Drude force field is very well optimized for α-helical peptides However, although it greatly outperforms the C36 force field, it performs slightly worse than the non-polarizable C22* force field in describing GB1 β-hairpin folding Using pertubations to the WHAM histograms, we showed that small enhancements to backbone polarizabilities improve both backbone hydration and β-hairpin folding while maintaining α-helical folding While the Drude model is a relatively cheap method to introduce dynamic, inducible atomic polarization, the cheaper non-polarizable C22* force field is sufficient for modeling β-sheets

Acknowledgements

Chris Rowley Benoit Roux Gumbart lab Funding Computational resources

Questions?